Practical  Carpentry 


WITH 


Steel  Square  Supplement 


ILLUSTRATED  EY  NEARLY  300  ENGRAVINGS. 


Practical  Carpentry 

WITH  STEEL  SQUARE  SUPPLEMENT 

BEING  A GUIDE  TO  THE 

Correct  Working  and  Laying  Out  of  All  Kinds  of 
Carpenters’  and  Joiners’  Work. 


With  the  Solutions  of  the  Various  Problems  in  Hip-Roofs, 
Gothic  Work,  Centering,  Splayed  Work,  Joints  and 
Jointing,  Hinging,  Dovetailing,  Mitering,  Timber 
Splicing,  Hopper  Work,  Skylights,  Raking 
Mouldings,  Circular  Work,  Etc.,  Etc. 

TO  WHICH  IS  PREFIXED  A THOROUGH  TREATISE  ON 

“CARPENTERS'  GEOMETRY." 


BY 

FRED  T.  HODGSON 

i.UTHOR  of  “The  Steel  Square  and  Its  Uses,  “The  Builder’s  Guide 
and  Estimator’s  Price  Book,”  “The  Slide  Rule  and 
How  to  Use  It,”  Etc.,  Etc. 


MONTGOMERY  WARD  & COMPANY 

CHICAGO 


TABLE  OF  CONTENTS. 


PART  I. 

PAG£ 

Geometry.— Straight  Lines.— Curved  Lines.— Solids.— Compound  Lines. 
—Parallel  Lines.— Oblique  or  Converging  Lines.— Plane  Figures.— 
Angles.— Right  Angles.— Acute  Angles.— Obtuse  Angles.— Right- 
angled  Triangles.— Quadrilateral  Figu res.— Paral lelograms.— Rect- 
angles.—Squares.— Rhomboids.—  Trapeziums.  —Trapezoids.— Diag- 
onals. —Polygons.—  Pentagons.  — Hexagons.  — Heptagons.  — Octa- 
gons.—Circles.— Chords.— Tangents.— Sectors.— Quadrants.— Arcs.— 
Concentric  and  Eccentric  Circles.— Altitudes.— Problems  I.  to  XXIX. 
—Drawing  of  Angles.— Construction  of  Geometrical  Figures.— Bisec- 
tion of  Lines.— Trisection  of  Lines  and  Angles.— Division  of  Lines 
into  any  Number  of  Parts.— Construction  of  Triangles.  Squares  and 
Parallelograms.— Construction  of  Proportionate  Squares.— Con- 
struction of  Polygons.— Areas  of  Polygons.— Areas  of  Concentric 
Rings  and  Circles.— Segments  of  Circles.— The  use  of  Ordinates 
for  Obtaining  Arcs  of  Circles.— Drawing  an  Ellipse  with  a Trammel. 
—Drawing  an  Ellipse  by  means  of  a String— Same  by  Ordinates.— 
Raking  Ellipses.— Ovals.— Sixty-two  Illustrations,  ...  $-34 


PART  n 

Arches,  Centres.— Window  and  Door  Heads.— Semi-circular  Arch.— 
Segmental  Arches.— Stilted  Arches.— Horseshoe  Arch.— Lancet  Arch. 
—Equilateral  Arch.— Gothic  Tracery.— Wheel-Windows.— Equila- 
teral Tracery.— Square  Tracery.— Finished  Leaf  Tracery.— Twenty- 
two  Illustrations.  • 


35-42 


VI 


TABLE  OF  CONTENTS. 


PART  III. 

PAGE 

Roofs.— Saddle  Roof.— Lean-to  or  Shed  Roof.— Simple  *Iip-Roof.— 
Pyramidal  Roof.— Theoretical  Roof.— Roof  with  Straining  Beam- 
Gothic  Roof.— Hammer-Beam  Roofs— Curved  Principal  Roofs.— 
Roofs  with  Suspending  Rods.— Deck  Roofs.— King-post  and  Prin- 
cipal Roof.— Queen-post  and  Principal  Roof.— Roofs  with  Laminated 
Arches.— Strapped  Roof  Frames.— Tie-beam  Roofs.— Roofs  for  Long 
Spans.— Theatre  Roof.— Church  Roof.— Mansard  Roof.— Slopes  of 
Roofs.— Rules  for  Determining  the  Sizes  of  Timbers  for  Roofs.— 
Acute  and  Obtuse  Angled  Hip-Roofs.— Development  of  Hip-Roofs.— 
Obtaining  Lengths  and  Bevels  of  Rafters.— Backing  Hip-Rafters.— 
Lengths,  Bevels  and  Cuts  of  Purlins.— Circular,  Conical  and  Seg- 
mental Roofs.— Rafters  with  Variable  Curves.— Veranda  Rafters.— 
Development  of  all  kinds  of  Rafters.— Curved  Mansard  Rafters.— 
Framed  Mansard  Roofs.— Lines  and  Rules  for  obtaining  various 
kinds  of  Information.— Thirty-four  Illustrations,  - - - 43— 


PART  IV. 

Covering  of  Roofs.— Shingling  Common  Roofs.— Shingling  Hip- 
Roofs.— Method  of  Shingling  on  Hip  Corner.— Covering  Circular 
Roofs.— Covering  Ellipsoidal  Roofs.— Valley  Roofs.— Four  Illustra- 
tions. ----------  65—68 


PART  V. 

The  Mitering  and  Adjusting  of  Mouldings.— Mitering  of  Spring 
Mouldings.— Preparing  the  Mitre-box  for  Cutting  Spring  Mould- 
ings.—Rules  for  Cutting  Mouldings,  with  Diagrams.— Mitre-boxes 
of  various  forms.— Lines  for  Spring  Mouldings  of  various  kinds.— 

Seven  Illustrations.  - 60—73 


PART  VI. 

Sashes  and  Skylights.— Raised  Skylights.— Skylights  with  Hips.— 
Octagon  Skylights  with  Segmental  Ribs.— Angle-bars,  with  Rules 
and  Diagrams,  showing  how  to  obtain  the  Angles,  Forms,  etc.— Sash- 
Bars,  Hints  on  their  Construction.— Twelve  Illustrations.  - - 74  78 


TABLE  OF  CONTENTS. 


V/ 


PAET  VII. 

PAGE 

Mouldings.— Angle  Brackets.— Corner  Coves.— Enlarging  and  Reduc- 
ing Mouldings.— Irregular  Mouldings.— Raking  Mouldings,  with 
Rules  for  Obtaining. — Mouldings  for  Plinths  and  Capitals  of  Gothic 
Columns. — Mouldings  around  Square  Standards. — Mitering  Cir- 
cular Mouldings  with  each  other  — Mitering  Circular  Mouldings 
with  Straight  ones.— Mitering  Mouldings  at  a Tangent.— Mitering 
Spring  Circular  Mouldings.— Description  of  Spring  Mouldings.— 

Lines  for  Circular  Spring  Mouldings.— Seventeen  Illustrations,  - 79—67 


PART  VIII. 

Joinery.— Dovetailing.— Common  Dovetailing.— Lapped  Dovetailing.— 

Blind  Dovetailing.— Square  Dovetailing.— Splay  Dovetailing.— 
Regular  and  Irregular  Dovetailing.— Lines  and  Cuts  for  Hoppers 
and  Splayed  Work.— Angles  and  Mitres  for  Splayed  Work.— Nineteen 
Illustrations.  - --  --  --  --  88—94 


PART  IX. 

Miscellaneous  Problems.— Bent  Work  for  Splayed  Jambs.— Develop 
ment  of  Cylinders.— Rules  and  Diagrams  for  Taking  Dimensions.— 
Angular  and  Curved  Measurements.— Eight  Illustrations.  - - 96—99 


PART  X. 

Joints  and  Straps.— Mortise  and  Tenon  Joints.— Toggle  Joints.— Hoot 
Joints.— Tongue  Joint.— Lap  Splice.— Scarfing.— Wedge  Joints.- 
King-bolts.— Straps,  Iron  Ties,  Sockets,  Bearing-plates,  Rings, 
Swivels  and  other  Iron  Fastenings.— Straining  Timbers,  Struts  and 
King- pieces.— Three  Plates,  Sixty-five  Illustrations,  - - 09-  JO) 


PART  XI. 

Hinging  and  Swing  Joints.— Door  Hinging.— Centre-pin  Hinging.— 

Blind  Hinging.— Folding  Hinging.— Knuckle  Hinging.— Pew  Hing- 
ing.—Window  Hinging.— Half-turn  Hinge.— Full-turn  Hinge.— Back 
Flap  Hinging.— Rule-joint.  Hinging.— Rebate  Hinging.— Three 
Plates.  Fifty-one  Illustrations.  - 102— 1C2 


TABLE  OF  CONTENTS. 


vm 

PART  XII. 

PAGE 

Useful  Rules  and  Tables.— Hints  on  the  Construction  of  Centres.— 

Rules  for  Estimating.— Form  of  Estimate.— Items  for  Estimating,— 
Remarks  on  Fences.— Nails:  sizes,  weights,  lengths  and  numbers.— 
Cornices,  Proportions  and  Projections  for  Different  Styles  of  Archi- 
tecture; and  Tall  and  Low  Buildings,  Verandas,  Bay  Windows  and 
Porches.— Proportion  of  Base-boards,  Dados,  Wainscots  and  Sur- 
bases.— Woods,  Hard  and  Soft,  their  Preparation,  and  how  to 
Finish.— Strength  and  Resistance  of  Timber  of  various  kinds.— 

Rules,  showing  Weight  and  other  qualities  of  Wood  and  Timber.— 

Stairs,  Width  of  Treads  and  Risers ; their  Cost;  how  to  Estimate  on 
them,  etc.— Inclinations  oj;  Roofs.— Contents  of  Boxes.  Bins  and 
Barrels.— Arithmetical  Signs.— Mensuration  of  Superficies.— Areas 
of  Squares,  Triangles,  Circles,  Regular  and  Irregular  Polygons.— 
Properties  of  Circles.— Solid  Bodies.— Gunter’s  Chain.— Drawing 
and  Drawing  Instruments.— Coloring  Drawings.— Coloring  for 
Various  Building  Materials.— Drawing  Papers.— Sizes  of  Drawing 
Papers.— Table  of  Board  Measure.— Nautical  Table.— Measure  of 
Time.— Authorized  Metric  System.— Measures  of  Length.— Mea- 
sures of  Surfaces.— Measures  of  Capacity.— Weights.— American 
Weights  and  Measures.— Square  Measure.— Cubic  Measure.— Cir- 
cular Measure.— Decimal  Approximations.— Form  of  Building 
Contract, - J04— Hi 


PRACTICAL  CARPENTRY 


PAST  I.— GEOMETRY. 


i^jlEFORE  a knowledge  of  geometry  can  be  acquired,  it  wiH 
K55  be  necessary  to  become  acquainted  with  some  of  the 
terms  and  definitions  used  in  the  science  of  geometry, 
and  to  this  end  the  following  terms  and  explanations  are  given, 
though  it  must  be  understood  that  these  are  only  a few  of  the 
terms  used  in  the  science,  but  they  are  sufficient  for  our  purposes : 

1.  A.  Point  has  position  but  not  magnitude.  Practically,  it  is 
represented  by  the  smallest  visible  mark  or  dot,  but  geometrically 
understood,  it  occupies  no  space.  The  extremities  or  ends  of  lines 
are  points ; and  when  two  or  more  lines  cross  one  another,  the 
places  that  mark  their  intersections  are  also  points. 

2.  A Line  has  length,  without  breadth  or  thickness,  and,  conse- 
quently, a true  geometrical  line  cannot  be  exhibited;  for  however 
finely  a line  may  be  drawn,  it  will  always  occupy  a certain  extent 
of  space. 

3.  A Superficies  or  Surface  has  length  and  breadth,  but  no  thick- 
ness. For  instance,  a shadow  gives  a very  good  representation  of 
a superficies : its  length  and  breadth  can  be  measured ; but  it  has 
no  depth  or  substance.  The  quantity  of  space  contained  in  any 
plane  surface  is  called  its  area. 

4.  A Plane  Superficies  is  a flat  surface,  which  will  coincide  with 
a straight  line  in  every  direction 


IO 


PRACTICAL  CARPENTRY. 


5.  A Curved  Superficies  is  an  uneven  surface,  or  such  as  will  not 
coincide  with  a straight  line  in  all  directions.  By  the  term  surface 
is  generally  understood  the  outside,  of  any  body  or  object;  as,  for 
instance,  the  exterior  of  a brick  or  stone,  the  boundaries  of  which 
are  represented  by  lines,  either  straight  or  curved,  according  to  the 
form  of  the  object.  We  must  always  bear  in  mind,  however,  that 
the  lines  thus  bounding  the  figure  occupy  no  part  of  the  surface ; 
hence  the  lines  or  points  traced  or  marked  on  any  body  or 
surface,  are  merely  symbols  of  the  true  geometrical  lines  or 
points. 

6.  A Solid  is  anything  which  has  length,  breadth  and  thickness; 
consequently,  the  term  may  be  applied  to  any  visible  object  con- 
taining substance ; but,  practically,  it  is  understood  to  signify  the 
solid  contents  or  measurement  contained  within  the  different  sur- 
faces of  which  any  body  is  formed. 

7.  Lmes  may  be  drawn  in  any  direction,  and  are  termed  straight, 
curved,  mixed,  concave,  or  convex  lines,  according  as  they  corres- 
pond to  the  following  definitions. 

8.  A Straight  Line  is  one  every  part  of  which  A“ — D 

lies  in  the  same  direction  between  its  extremities,  Flg'  *' 
and  is,  of  course,  the  shortest  distance  between  two  points,  as 
from  a to  b,  Fig.  1. 

9.  A Curved  Liiie  is  such  that  it  does  not  lie  in  a straight  direc- 
tion between  its  extremities,  but  is  continually  changing  by  inflec- 
tion. It  may  be  either  regular  or  irregular. 

10.  A Mixed  or  Compound  Line  is  composed  of  straight  and 
curved  lines,  connected  in  any  form. 

11.  A Concave  or  Cojivex  Lme  is  such  that  it  cannot  be  cut  by 
a straight  line  in  more  than  two  points ; the  concave  or  hollow 
side  is  turned  towards  the  straight  line,  while  the  convex  or  swell- 
ing side  looks  away  from  it.  For  instance,  the  inside  of  a basin  is 
concave — the  outside  of  a ball  is  convex. 

12.  Parallel  Straight  Lines  have  no  inclination,  but  are  every- 
where at  an  equal  distance  from  each  other;  consequently  they 
can  never  meet,  though  produced  or  continued  to  infinity  in  either 
or  both  directions.  Parallel  lines  may  be  either  straight  or  curved, 


PRACTICAL  CARPENTRY. 


II 


provided  they  are  equally  distant  from  each  other  throughout  their 
extension. 

13.  Oblique  or  Converging  Lines  are  straight  lines,  which,  if  con- 
tinued, being  in  the  same  plane,  change  their  distance  so  as  to 
meet  or  intersect  each  other. 

14.  A Plane  Figure , Scheme , or  Diagram , is  the  lineal  representa- 
tion of  any  object  on  a plane  surface.  If  it  is  bounded  by  straight 
lines,  it  is  called  a rectilineal  figure;  and  if  by  curved  lines,  a 
curvilineal  figure. 

15.  An  Angle  is  formed  by  the  inclination  of  two  lines  meeting 
in  a point:  the  lines  thus  forming  the  angle  are  called  the  sides; 
and  the  point  where  the  lines  meet  is  called  the  vertex  or  angular 
point . 

When  an  angle  is  expressed  by  three  letters,  as  a b c,  Fig.  2,  the 
middle  letter  b should  always  denote  the  angular  point:  where 


Fig.  2- 


there  is  only  one  angle,  it  may  be  expressed  more  concisely  by  a 
letter  placed  at  the  angular  point  only,  as  the  angle  at  a,  Fig.  3. 

16.  The  quantity  of  an  angle  is  estimated  by  the  arc  of  any 
circle  contained  between  the  two  sides  or  lines  forming  the  angle; 
the  junction  of  the  two  lines,  or  vertex  of  the  angle,  being  the 
centre  from  which  the  arc  is  described.  As  the  circumferences  of 
all  circles  are  proportional  to  their  diameters,  the  arcs  of  similar 
sectors  also  bear  the  same  proportion  to  their  respective  circum- 
ferences; and,  consequently,  are  proportional  to  their  diameters, 
and,  of  course,  also  to  their  radii  or  semi-diameters>  Hence,  the 


12 


PRACTICAL  CARPENTRY. 


proportion  which  the  arc  of  any  circle  bears  to  the  circumference 
of  that  circle,  determines  the  magnitude  of  the  angle.  From  this 
it  is  evident  that  the  quantity  or  magnitude  of  angles  does  not  de- 
pend upon  the  length  of  the  sides  or  radii  forming  them,  but 
wholly  upon  the  number  of  degrees  contained  in  the  arc  cut  from 
the  circumference  of  the  circle  by  the  opening  of  these  lines.  The 
circumference  of  every  circle  is  divided  by  mathematicians  into 
360  equal  parts,  called  degrees ; each  degree  being  again  subdi- 
vided into  60  equal  parts,  called  minutes,  and  each  minute  into  60 
parts,  called  seconds.  Hence,  it  follows  that  the  arc  of  a quarter 
circle  or  quadrant  includes  90  degrees ; that  is,  one-fourth  part  of 
360  degrees.  By  dividing  a quarter  circle,  that  is,  the  portion  of 
the  circumference  of  any  circle  contained  between  two  radii  form- 
ing a right  angle,  into  90  equal  parts,  or,  as  is  shown  in  Fig.  4,  into 
nine  equal  parts  of  10  degrees  each,  then  drawing  straight  lines 
from  the  centre  through  each  point  of  division  in  the  arc;  the  right 
angle  will  be  divided  into  nine  equal  angles,  each  containing  10 
degrees.  Thus,  suppose  b c the  horizontal  line,  and  a b the  per- 
pendicular ascending  from  it,  any  line  drawn  from  b — the  centre 
from  which  the  arc  is  described — to  any  point  in  its  circumference, 
determines  the  degree  of  inclination  or  angle  formed  between  it 
and  the  horizontal  line  b c.  Thus,  a line  from  the  centre  b to  the 
tenth  degree,  separates  an  angle  of  10  degrees,  and  so  on.  In  this 
manner  the  various  slopes  or  inclinations  of  angles  are  defined. 

17.  A Right  Angle  is  produced  by  one  straight  line  standing 
upon  another,  so  as  to  make  the  adjacent  angles  equal.  This  is 
what  workmen  call  “ square,”  and  is  the  most  useful  figure  they 
employ. 

18.  An  Acute  Angle  is  less  than  a right  angle,  or  less  than  90 
degrees. 

19.  An  Obtuse  Angle  is  greater  than  a right  angle  or  square,  or 
more  than  90  degrees. 

The  number  of  degrees  by  which  an  angle  is  less  than  90  de- 
grees is  called  the  complement  of  the  angle.  Also,  the  difference 
between  an  obtuse  angle  and  a semicircle,  or  180  degrees,  is  called 
the  supplement  of  that  angle. 


PRACTICAL  CARPENTRY. 


l3 


20.  Plane  Figures  are  bounded  by  straight  lines,  and  are  named 
according  to  the  number  of  sides  which  they  contain.  Thus,  the 
space  included  within  three  straight  lines,  forming  three  angles, 
is  called  a trilateral  figure  or  triangle. 

21.  A Right-Angled*  Triangle  has  one  right  angle:  the  sides 
forming  the  right  angle  are  called  the  base  and  perpendicular;  and 
the  side  opposite  the  right  angle  is  named  the  hypothenuse.  An 
equilateral  triangle  has  all  its  sides  of  equal  length.  An  isosceles 
triangle  has  only  two  sides  equal;  a scalene  triangle  has  all  its 
sides  unequal.  An  acute-angled  triangle  has  all  its  angles  acute, 
and  an  obtused-angled  triangle  has  one  of  its  angles  only 
obtuse. 

The  triangle  is  one  of  the  most  useful  geometrical  figures  for  the 
mechanic  in  taking  dimensions;  for  since  all  figures  that  arc 
bounded  by  straight  lines  are  capable  of  being  divided  into  tri- 
angles, and  as  the  form  of  a triangle  cannot  be  altered  without 
changing  the  length  of  some  one  of  its  sides,  it  follows  that  the  true 
form  of  any  figure  can  be  preserved  if  the  length  of  the  sides  of 
the  different  triangles  into  which  it  is  divided  is  known ; and  the 
area  of  any  triangle  can  easily  be  ascertained  by  the  same  rule,  as 
will  be  shown  further  on. 

Quadrilateral  Figures  are  literally  four- sided  figures.  They  are 
also  called  quadrangles,  because  they  have  four  angles. 

22.  A Parallelogram  is  a figure  whose  opposite  sides  are  parallel, 
as  a b c d,  Fig.  5. 

23.  A Rectangle  is  a parallelogram  having  four  right  angles,  as  a 
b c d,  in  Fig.  5. 

24.  A Square  is  an  equilateral  rectangle,  having  all  its  sides 
equal,  like  Fig.  5. 

25.  An  Oblong  is  a rectangle  whose  adjacent  sides  are  unequal, 
as  the  parallelogram  shown  at  Fig.  10. 

26.  A Rhombus  is  an  oblique-angled  figure,  or  parallelogram 
having  four  equal  sides,  whose  opposite  angles  only  are  equal,  as 
C,  Fig.  6. 

27.  A Rhomboid  is  an  oblique-angled  parallelogram,  of  which  ths, 
adjoining  sides  are  unequal,  as  d,  Fig.  7. 


H 


PRACTICAL  CARPENTRY. 


28.  A Trapezium  is  an  irregular  quadrilateral  figure,  having  no 
two  sides  parallel,  as  e,  Fig.  8. 

29.  A Trapezoid  is  a quadrilateral  figure,  which  has  two  of  its 
opposite  sides  parallel,  and  the  remaining  two  neither  parallel  nor 
equal  to  one  another,  as  f,  Fig.  9. 


Fig.  8.  Fig.  9. 


30.  A Diagonal  is  a straight  line  drawn  between  two  opposite 
angular  points  of  a quadrilateral  figure,  or  between  any  two  angular 
points  of  a polygon.  Should  the  figure  be  a parallelogram,  the 
diagonal  will  divide  it  into  two  equal  triangles,  the  opposite  sides 
and  angles  of  which  will  be  equal  to  one  another.  Let  abcd, 
Fig.  10,  be  a parallelogram;  join  a c,  then  a c is  a diagonal,  and 
the  triangles  adc,abc,  into  which  it  divides  the  parallelogram, 
are  equal. 

31.  A plane  figure,  bounded  by  more  than  four  straight  lines,  is 
called  a Polygon . A regular  polygon  has  all  its  sides  equal,  and 
consequently  its  angles  are  also  equal,  as  k,  l,  m,  and  n,  Figs. 


12-15.  An  irregular  polygon  has  its  sides  and  angles  unequal,  as 
H,  Fig.  11.  Polygons  are  named  according  to  the  number  of  their 
sides  or  angles,  as  follows 

32.  A Pentago?i  is  a polygon  of  five  sides,  as  h or  k,  Figs.  11,  12. 

33.  A Hexagon  is  a polygon  of  six  sides,  as  l,  Fig.  13. 

34.  A., Heptagon  has  seven  sides,  as  m.  Fig.  14. 


PRACTICAL  CARPENTRY. 


1 s 


35.  An  Octagon  has  eight  sides,  as  n,  Fig.  15. 

An  Enneagoti  has  nine,  a Decagon  ten,  an  Undecagon  eleven,  and 
a Dodecagon  twelve  sides.  Figures  having  more  than  twelve  sides 
are  generally  designated  Polygons , or  many-angled  figures. 

36.  A Circle  is  a plane  figure  bounded  by  one  uniformly  curved 
line,  bed  (Fig.  16),  called  the  circumference,  every  part  of  which 
is  equally  distant  from  a point  within  it,  called  the  centre,  as  a . 

37.  The  Radius  of  a circle  is  a straight  line  drawn  from  the 
centre  to  the  circumference ; hence,  all  the  radii  (plural  for  radius) 
of  the  same  circle  are  equal,  as  b ay  c a,  e a,f  a,  in  Fig.  16. 

38.  The  Diameter  of  a circle  is  a straight  line  drawn  through  the 
centre,  and  terminated  on  each  side  by  the  circumference;  conse- 


quently the  diameter  is  exactly  twice  the  length  of  the  radius ; and 
hence  the  radius  is  sometimes  called  the  semi-diameter.  (See  baet 
Fig.  16.) 

49.  The  Chord  or  Subtens  of  an  arc  is  any  straight  line  drawn 
from  one  point  in  the  circumference  of  a circle  to  another,  joining 
the  extremities  of  the  arc,  and  dividing  the  circle  either  into  two 
equal,  or  two  unequal  parts.  If  into  equal  parts,  the  chord  is  also 
the  diameter,  and  the  space  included  between  the  arc  and  the  di- 
ameter, on  either  side  of  it,  is  called  a semicircle,  as  baem  Fig.  16. 
If  the  parts  cut  off  by  the  chord  are  unequal,  each  of  them  is  called 
a segme?it  of  the  circle.  The  same  chord  is  therefore  common  to 
two  arcs  and  two  segments ; but,  unless  when  stated  otherwise,  it 
is  always  understood  that  the  lesser  arc  or  segment  is  spoken  of,  as 
in  Fig.  16,  the  chord  c d is  the  chord  of  the  arc  c e d. 

If  a straight  line  be  drawn  from  the  centre  of  a circle  to  meet 
the  chord  of  an  arc  perpendicularly,  as  a / in  Fig.  16,  it  will  divide 
the  chord  into  two  equal  parts,  and  if  the  straight  line  be  produced 


? 6 


PRACTICAL  CARPENTRY. 


to  meet  the  arc,  it  will  also  divide  it  into  two  equal  parts,  as 

cfjd. 

Each  half  of  the  chord  is  called  the  sine  of  the  half-arc  to  which 
it  is  opposite ; and  the  line  drawn  from  the  centre  to  meet  the 
chord  perpendicularly,  is  called  the  co-sine  of  the  half-arc.  Con- 
sequently, the  radius,  the  sine,  and  co-sine  of  an  arc  form  a right 
angle. 

40.  Any  line  which  cuts  the  circumference  in  two  points,  or  a 
chord  lengthened  out  so  as  to  extend  beyond  the  boundaries  of 
the  circle,  such  as  g h in  Fig.  17,  is  sometimes  called  a Secant . 
But,  in  trigonometry,  the  secant  is  a line  drawn  from  the  centre 
through  one  extremity  of  the  arc,  so  as  to  meet  the  tangent  which  is 
drawn  from  the  other  extremity  at  right  angles  to  the  radius. 
Thus,  Feb  is  the  secant  of  the  arc  c e , or  the  angle  cf  e,\n  Fig.  17 

41.  A Tangent  is  any  straight  line  which  touches  the  circumfer- 
ence of  a circle  in  one  point,  which  is  called  the  point  of  contact, 
as  in  the  tangent  line  e b , Fig.  17. 

42.  A Sector  is  the  space  included  between  any  two  radii,  and 
that  portion  of  the  circumference  comprised  between  them:  ce  f 
is  a sector  of  the  circle  a f c e , Fig.  17. 

43.  A Quadrant , or  quarter  of  a circle,  is  a sector  bounded  by 
two  radii,  forming  a right  angle  at  the  centre,  and  having  one- 
fourth  part  of  the  circumference  for  its  arc,  as  f fd , Fig.  1 7. 

44.  An  Arc , or  Arch , is  any  portion  of  the  circumference  of  a 
circle,  as  c d e , Fig.  17. 

It  may  not  be  improper  to  remark  here  that  the  terms  circle  and 
circumference  are  frequently  misapplied.  Thus  we  say,  describe  a 
circle  from  a given  point,  etc.,  instead  of  saying  describe  the  cir- 
cumference of  a circle — the  circumference  being  the  curved  line 
thus  described,  everywhere  equally  distant  from  a point  within  it, 
called  the  centre : whereas  the  circle  is  properly  the  superficial 
space  included  within  that  circumference. 

45.  Concentric  Circles  are  circles  within  circles,  described  from 
the  same  centre;  consequently,  their  circumferences  are  parallel  to 
one  another,  as  Fig.  18. 

46.  Eccentric  Circles  are  those  which  are  not  described  from  the 


PRACTICAL  CARPENTRY* 


*7 


same  centre;  any  point  which  is  not  the  centre  is  also  eccentric  in 
reference  to  the  circumference  of  that  circle.  Eccentric  circles 
may  also  be  tangent  circles ; that  is,  such  as  come  in  contact  in 
one  point  only,  as  Fig.  19. 

47.  Altitude . The  height  of  a triangle  or  other  figure  is  called 
its  altitude . To  measure  the  altitude,  let  fall  a straight  line  from 
the  vertex,  or  highest  point  in  the  figure,  perpendicular  to  the 
base  or  opposite  side;  or  to  the  base  continued,  as  at  b d,  Fig.  20, 
should  the  form  of  the  figure  require  its  extension.  Thus  c d is  the 
altitude  of  the  triangle  abc. 

We  have  now  described  all  the  figures  we  shall  require  for  the 
purpose  of  thoroughly  understanding  all  that  will  follow  in  this 
book ; but  we  would  like  to  say  right  here  that  the  student  who 
has  time  should  not  stop  at  this  point  in  the  study  of  geometry,  for 
the  time  spent. in  obtaining  a thorough  knowledge  of  this  useful 


science  will  bring  in  better  returns  in  enjoyment  and  money,  than 
if  expended  for  any  other  purpose. 

We  will  now  proceed  to  explain  hew  the  figures  we  have  de- 
scribed can  be  constructed.  There  are  several  ways  of  constructing 
nearly  every  figure  we  produce,  but  we  have  chosen  those  methods 
that  seemed  to  us  the  best,  and  to  save  space  have  given  as  few 
examples  as  possible  consistent  with  efficiency. 

Problem  I. — Through  a given  point  c (Fig.  18  a),  to  draw  a 
straight  line  parallel  to  a given  straight  line  a b. 

In  a b (Fig.  18  a)  take  any  point  d , and  from  d as  a centre  with 
the  radius  d c,  describe  an  arc  c e , cutting  a b in  e,  and  from  c as  a 


i8 


PRACTICAL  CARPENTRY. 


centre,  with  the  same  radius,  describe  the  arc  */d,  make  dv>  equal  t 
c <?,  join  c d,  and  it  will  be  parallel  to  a b. 

Problem  II. — To  make  an  angle  equal  to  a given  rectilineal  angle . 

From  a given  point  e (Fig.  19  a),  upon  the  straight  line  e f,  to. 
make  an  angle  equal  to  the  given  angle  abc,  From  the  angulai 
point  b,  with  any  radius,  describe  the  arc*? /,  cutting  bc  and  b a in 
the  points  e and  /.  From  the  point  e on  e f with  the  same  radius, 


describe  the  arc  hg,  and  make  it  equal  to  the  arc  e /;  then  from  e, 
through  g , draw  the  line  e d : the  angle  d e f will  be  equal  to  the 
angle  abc. 

Problem  III. — To  bisect  a given  angle. 

Let  abc  (Fig.  20 -a)  be  the-  given  angle.  From  the  angular  point 
b,  with  any  radius,  describe  an  arc  cutting  b a and  b c in  the 
points  d and  e\  also,  from  the  points  d and  e as  centres,  with  any 
radius  greater  than  half  the  distance  between  them,  describe  arcs 
cutting  each  other  in  /;  through  the  points  of  intersection  f draw 
b/d  : the  angle  a b c is  bisected  by  the  straight  line  b d;  that  is, 
it  is  divided  into  two  equal  angles,  a b d'  and  cbd. 

Problem  IV. — To  trisect  or  divide  a right  angle  into  three  equal 
angles . 

Let  abc  (Fig.  21)  be  the  given  right  angle.  From  the  angular 


Fig.  J8a. 


Fig.  19  a. 


■0 


Fig.  20  a. 


PRACTICAL  CARPENTRY. 


T9 


point  B,  with  any  radius,  describe  an  arc  cutting  b a and  b c in  the 
points  d and  g;  from  the  points  d and^*,  with  the  radius  b d or  b g, 
describe  the  arcs^cutting  the  arc  d g me  and /;  join  b e and  b /: 
these  lines  will  trisect  the  angle  a b c,  or  divide  it  into  three  equal 
angles. 

The  trisection  of  an  angle  can  be  effected  by  means  of  elementary 
geometry  only  in  a very  few  cases;  such,  for  instance,  as  those 
where  the,  arc  which  measures  the  proposed  angle  is  a whole  circle, 
or  a half,  a fourth,  or  a fifth  part  of  the  circumference.  Any  angle 
of  a pentagon  is  trisected  by  diagonals,  drawn  to  its  opposite 
angles. 

Problem  V. — From  a give?i  point  c,  in  a given  straight  line  a b, 
to  erect  a perpendicular . 

From  the  point  c (Fig.  22),  with  any  radius  less  than  c a or  c b, 
describe  arcs  cutting  the  given  line  a b in  d and  e\  from  these 


/ 

K 


A 


A 


C 

Fig.  52- 


h 


points  as  centres,  with  a radius  greater  than  cd  or  c e,  describe  arcs 
intersecting  each  other  in  /:  join  c /,  and  this  line  will  be  the  per- 
pendicular required. 

Another  Method. — To  draw  a right  angle  or  erect  a perpendicular 
by  means  of  any  scale  of  equal  parts,  or  standard  measure  of  inches, 
feet,  yards,  etc.,  bv  setting  off  distances  in  proportion  to  the  num- 
bers 3,  4 and  5,  or  6,  8 and  10,  or  any  numbers  whose  squares  cor- 
respond to  the  sides  and  hypothenuse  of  a right-angled  triangle. 

From  any  scale  of  equal  parts,  as  that  represented  by  the  line  d 
(Fig.  23),  which  contains  5,  set  off  from  b,  on  the  line  a b,  the  dis- 
tance b e , equal  to  3 of  these  parts  ; then  from  b,  with  a radius 
equal  to  4 of  the  same  parts,  describe  the  arc  a b\  also  from  e as  a 


20 


PRACTICAL  CARPENTRY. 


\centre,  with  a radius  equal  to  5 parts,  describe  another  arc  inter- 
secting the  former  in  c;  lastly  join  b c;  the  line  b c will  be  per- 
pendicular to,A  B. 

This  mode,  of  drawing  fight  angles  is  more  troublesome  upon 
paper  than  the  method  previously  given  ; but  in  laying  out  grounds 
or  foundations  of  buildings  it  is  often  very  useful,  since  only  with  a 
ten-foot  pole,  tape  line,  or  chain,  perpendiculars  may  be  set  out 
very  accurately.  The  method  is  demonstrated  thus : — The  square 
of  the  hypothenuse,  or  longest  side  of  a right-angled  triangle,  being 
equal  to  the  sum  of  the  squares  of  the  other  two  sides,  the  same 
property  must  always  be  inherent  in  any  three  numbers,  of  which 


the  squares  of  the  two  lesser  numbers,  added  together,  are  equal  fo 
the  square  of  the  greater.  For  example,  take  the  numbers  3,  4, 
and  s ; the  square  of  3 is  9,  and  the  square  of  4 is  16;  16  and  9, 
added  together  make  25,  which  is  5 times  5,  or  the  square  of  the 
greater  number.  Although  these  numbers,  or  any  multiple  of 
them,  such  as  6,  8,  10,  or  12,  16,  20,  etc.,  are. the  most  simple,  and 
most  easily  retained  in  the  memory,  yet  there  are  other  numbers, 
very  different  in  proportion,  which  can  be  made  to  serve  the  same 
purpose.  Let  11  denote  any  number;  then  n2  + 1,  n2  — 1,  and  2;/, 
will  represent  the  hypothenuse,  base,  and  perpendicular  of  a right- 
angled  triangle.  Suppose  n = 6,  then  n2  + 1 = 37,  «2—  1 = 35^ 
and  2n  = 12 : hence,  37,  35,  and  12  are  the  sides  of  a right-angled 
triangle.  A knowledge  of  this  problem  will  often  prove  of  the 
greatest  service  to  the  workman. 


PRACTICAL  CARPENTRY. 


2 1 


Problem  VI. — To  bisect  a given  straight  line . 

Let  a b (Fig.  24)  be  the  given  straight  line.  From  the  extreme 
points  a and  b as  centres,  with  any  equal  radii  greater  than  half 
the  length  of  a b,  describe  arcs  cutting  each  other  in  c and  d : a 
straight  line  drawn  through  the  points  of  intersection  c and  d,  will 
bisect  the  line  a b in  e. 

Problem  VII. — To  divide  a given  line  into  any  number  of  equal 
parts . 

Let  a b (Fig.  25)  be  the  given  line  to  be  divided  into  five  equal 
parts.  From  the  point  a draw  the  straight  line  a c,  forming  any 
angle  with  a b.  On  the  line  a c,  with  any  convenient  opening  of 
the  compasses,  set  off  five  equal  parts  towards  c ; join  the  extreme 


Fig.  25.  Fig.  26  Fig.  27 


points  c b;  through  the  remaining  points  1,  2,  3,  and  4,  draw  lines 
parallel  to  b c,  cutting  ab  in  the  corresponding  points,  1 , 2,3, 
and  4 : ab  will  be  divided  into  five  equal  parts,  as  required. 

There  are  several  other  methods  by  which  lines  may  be  divided 
into  equal  parts ; they  are  not  necessary,  however,  for  our  purpose, 
so  we  will  content  ourselves  with  showing  how  this  problem  may 
be  used  for  changing  the  scales  of  drawings  whenever  such  change 
is  desired.  Let  a b (Fig.  26)  represent  the  length  of  one  scale  or 
drawing,  divided  into  the  given  parts  a d,  d e,  e ffg , gh,  and  h b; 
and  d e the  length  of  another  scale  or  drawing  required  to  be  di- 
vided into  similar  parts.  From  the  point  r draw  a line  b c = d e, 
and  forming  any  angle  with  a b;  join  a c,  and  through  the  points 
e>f  and  h,  drav(  d k,  e /,  /w,  g n , h 0 , parallel  to  a c ; and 
the  parts  c k,  k /,  l m,  etc.,  will  be  to  each  other,  or  to  the  whole 
line  b c,  as  the  lines  a d,  d e,  e f,  etc.,  are  to  each  other,  or  to  the 
given  line  or  scale  a b.  By  this  method,  as  will  be  evident  from 


22 


PRACTICAL  CARPENTRY. 


the  figure,  similar  divisions  can  be  obtained  in  lines  of  any  given 
length. 

Problem  VIII, — To  describe  ah  equilateral  triangle  lipon  a given 
straight  line. 

Let  a b (Fig.  27)  be  the  given  straight  line.  From  the  points 
a and  b,  with  a radius  equal  to  a b,  describe  arcs  intersecting  each 
other  in  the  point  c.  Join  c a and  c b,  and  abc  will  be  the 
equilateral  triangle  required. 

Problem  IX. — To  construct  a triangle  whose  sides  shall  be  equal 
to  three  given  lines , F,  E,  D. 

Draw  a B (Fig.  28)  equal  to  the  given  line  f.  From  a as  a 
centre,  with  a radius  equal  to  the  line  e,  describe  an  arc;  then 


c 


A 

E — 

F — 


from  B as  a centre,  with  a rad  ms  equal  to  the  line  d,  describe  an- 
other arc  intersecting  the  former  in  c;  join  c a and  c b,  and  abc 
will  Be  the  triangle  required. 

Problem  X. — To  describe  a rectangle  or  parallelogram  having  one 
of  its  sides  equal  to  a given  li?ie,  and  its  area  equal  to  that  of  a given 
rectangle . 

Let  a b (Fig.  29)  be  the  given  line,  and  c d e f the  given 
rectangle.  Produce  c e to  g,  making  e g equal  to  ab;  from  g 
draw  g k parallel  to  f f,  and  meeting  d f produced  in  11.  Draw 
the  diagonal  g f,  extending  it  to  meet  c d produced  in  l ; also 
draw  l k parallel  to  d h,  and  produce  e f till  it  meet  l k in  m ; 
then  f m k h is  the  rectangle  required. 


PRACTICAL  carpentry. 


23 


Equal  and  similar  rhomboids  or  parallelograms  of  any  dimen- 
sions may  be  drawn  after  the  same  manner,  seeing  the  comple- 
ments of  the  parallelograms  which  are  described  on  or  about  the 
diagonal  of  any  parallelogram,  are  always  equal  to  each  other; 
while  the  parallelograms  themselves  are  always  similar  to  each 
other,  and  to  the  original  parallelogram  about  the  diagonal  of 
which  they  are  constructed.  Thus,  in  the  parallelogram  c c k l 
the  complements  cefd  and  f m k h are  always  equal,  while  the 
parallelograms  efhg  and  d f m l about  the  diagonal  g l,  are 
always  similar  to  each  other,  and  to  the  whole  parallelogram 
CGKL 

Problem  XI. — To  describe  a square  equal  to  two  given  squares. 

Let  a and  b (Fig.  30)  be  the  given  squares.  Place  them  so  that 
a side  of  each  may  form  the  right  angle  d c e ; join  d e,  and  upon 
this  hypothenuse  describe  the  square  degf,  and  it  will  be  equal 
to  the  sum  of  the  squares  a and  b,  which  are 
constructed  upon  the  legs  of  the  right-angled 
triangle  dce.  In  the  same  manner,  any 
other  rectilineal  figure,  or  even  circle,  may 
be  found  equal  to  the  sum  of  other  two  simi- 
lar figures  or  circles.  Suppose  the  lines  c d 
and  c e to  be  the  diameters  of  two  circles, 
then  d e will  be  the  diameter  of  a third, 
equal  in  area  to  the  other  two  circles.  Or 
suppose  c d and  c e to  be  the  like  sides  of 
any  two  similar  figures,  then  d e will  be  the 
corresponding  side  of  another  similar  figure 
equal  to  both  the  former. 

Problem  XII. — To  describe  a square  equal  to  any  number  of  given 
squares. 

Let  it  be  required  to  construct  a square  equal  to  the  three  given 
squares  a,  b,  and  c (Fig.  31).  Take  the  line  d e,  equal  to  the  side 
of  the  square  c.  From  the  extremity  d erect  d f perpendicular  to 
d e,  and  equal  to  the  side  of  the  square  b ; join  e f;  then  a square 
described  upon  this  line  will  be  equal  to  the  sum  of  the  two  given 
squares  c and  b.  Again,  upon  the  straight  line  e f erect  the  per- 


24  PRACTICAL  CARPENTRY. 

pendicular  f g,  equal  to  the  side  of  the 
third  given  square  a ; and  join  g e,  which 
will  be  the  side  of  the  square  g e h k, 
equal  in  area  to  a,  b,  and  c.  Proceed  in 
the  same  way  for  any  number  of  given 
squares. 

Problem  XIII. — Upon  a given  straight 
line  to  describe  a regular  polygon . 

To  produce  a regular  pentagon  draw 
A b to  c (Fig.  32),  so  that  b c may  be 
equal  to  a b ; from  b as  a centre,  with  the 
radius  b a or  b c describe  the  semicircle 
A d c;  divide  the  semi-circumference 
adc  into  as  many  equal  parts  as  there 
are  parts  in  the  required  polygon,  which, 
in  the  present  case,  will  be  five ; through  the  second  division  from 
c draw  the  straight  line  b d,  which  will  form  another  side  of  the 
figure.  Bisect  a b at  e>  and  bd  at  f and  draw  e g and  / G per- 
pendicular to  a b and  b d ; then  g,  the 
point  of  intersection,  is  the  centre  of  a 
circle,  of  which  a b and  d are  points 
in  the  circumference.  From  g,  with  a 
radius  equal  to  its  distance  from  any  of 
these  points,  describe  the  circumference 
a b d h k ; then  producing  the  dotted 
lines  from  the  centre  b,  through  the 
remaining  divisions  in  the  semicircle 
a d c,  so  as  to  meet  the  circumference  of  which  g is  the  centre, 
in  h and  k,  these  points  will'  divide  the  circle  a b d h k into  the 
number  of  parts  required,  each  part  being  equal  to  the  given  side 
of  the  pentagon. 

From  the  preceding  example  it  is  evident  that  polygons  of  any 
number  of  sides  may  be  constructed  upon  the  same  principles,  be- 
cause the  circumferences  of  all  circles,  when  divided  into  the  same 
number  of  equal  parts,  produce  equal  angles ; and,  consequently, 
by  dividing  the  semi-circumference  of  any  given  circle  into  the 


H 


D G 


PRACTICAL  CARPENTRY. 


*S 


number  of  parts  required,  two  of  these  parts  will  form  an  angle 
which  will  be  subtended  by  its  corresponding  part  of  the  whole 
circumference.  And  as  all  regular  polygons  can  be  inscribed  in  a 
circle,  it  must  necessarily  follow,  that  if  a circle  be  described 
through  three  given  angles  of  that  polygon,  it  will  contain  the 
number  of  sides  or  angles  required. 

The  above  is  a general  rule,  by  which  all  regular  polygons  may 
be  described  upon  a given  straight  line;  but  there  are  other 
methods  by  which  many  of  them  may  be  more  expeditiously  con- 
structed, as  shown  in  the  following  examples : — 

Problem  XIV. — Upon  a given  straight  line  to  describe  a regular 
pentagon . 

Let  a b (Fig.  33)  be  the  given  straight  line;  from  its  extremity 
B erect  Be  perpendicular  to  a b,  and  equal 
to  its  half.  Join  a c , and  produce  it  till 
c d be  equal  to  b c , or  half  the  given  line 
a b.  From  a and  b as  centres,  with  a 
radius  equal  to  b d,  describe  arcs  inter- 
secting each  other  in  e,  which  will  be  the 
centre  of  the  circumscribing  circle  abf 
g h.  The  side  a b applied  successively 
to  this  circumference,  will  give  the  angu- 
lar points  of  the  pentagon;  and  these 
being  connected  by  straight  lines  will  complete  the  figure. 

Problem  XV. — Upon  a given  straight  line  to  describe  a regular 
hexagon . — 

Let  a b (Fig.  34)  be  the  given  straight  line.  From  the  extremi- 
ties a and  b as  centres,  with  the  radius  a b describe  arcs  cutting 
each  other  in  g.  Again  from  g,  the  point  of  intersection,  with  the 
same  radius,  describe  the  circle  abc,  which  will  contain  the  given 
side  a b six  times  when  applied  to  its  circumference,  and  will  he 
the  hexagon  required. 

Problem  XVI. — To  describe  a regular  octagon  upon  a given 
straight  line. 

Let  a b (Fig.  35)  be  the  given  line.  From  the  extremities  a 
and  b erect  the  perpendiculars  a e and  b f ; extend  the  given 


PRACTICAL  CARPENVRY. 


26 

line  both  ways  to  k and  /,  forming  external  right  angles  with  the 
lines  a e and  b f.  Bisect  these  external  right  angles,  making  each 
of  the  bisecting  lines  a h and  b c equal  to  the  given  line  a b. 
Draw  h G and  c d parallel  to  a e or  b f,  and  each  equal  in  length 
to  a b.  From  g draw  g e parallel  to  b c,  and  intersecting  a e in 
e,  and  from  d draw  d f parallel  to  a h,  intersecting  b f in  f. 
Join  e f,  and  abcdfeghIs  the  octagon  required.  Or  from  d 


and  g as  centres,  with  the  given  line  a b as  radius,  describe  arcs 
cutting  the  perpendiculars  a e and  b f in  e and  f,  and  join  g e, 
e f,  f d,  to  complete  the  octagon. 

Otherwise , thus. — Let  a b (Fig.  36)  be  the  given  straight  line  on 
which  the  octagon  is  to  be  described.  Bisect  it  in  a , and  draw 
the  perpendicular  'a  b equal  to  a a or  b a.  Join  a b , and  produce 
a b to  c,  making  b c equal  to  a b ; join  also  a e and  b c , extending 
them  so  as  to  make  c e and  c f each  equal  to  a c or  b c. 
Through  c draw  c c g at  right  angles  to  a e.  Again,  through  the 

same  point  c . draw  d h at  right  angles  to  b f,  making  each  of  the 

lines  fC,  f D.  c g,  and  c h equal  to  a c or  c b,  and  consequently 
equal  to  one  another.  Lastly,  join  bc,cd,de,ef,fg,gh,ha; 
abcdefgh  will  be  a regular  octagon  described  upon  a b,  as 
required. 

Problem  XVII. — In  a given  square  to  inscribe  a given  octagon . 

Let  a b c d (Fig.  37)  be  the  given  square.  Draw  the  diagonals 
a c and  b d,  intersecting  each  other  in  e ; then  from  the  angular 
points  abc  and  d as  centres,  with  a radius  equal  to  half  the 

diagonal,  viz.,  a e or  c <?,  describe  arcs  cutting  the  sides  of  tht* 


PRACTICAL  CARPENTRY. 


27 


square  in  the  points  f g , h , k , /,  m,  n , <?,  and  the  straight  lines  0 
g h,  k l,  and  m n , joining  these  points  will  complete  the  octagon, 
and  be  inscribed  in  the  square  a b c d,  as  required. 

Problem  XVIII. — To  find  the  area  of  any  regular  polygon . 

Let  the  given  figure  be  a hexagon;  it  is  required  to  find  its  area. 
Bisect  any  two  adjacent  angles,  as  those  at  a and  b (Fig.  38),  by 
the  straight  lines  a c and  b c,  intersecting  in  c,  which  will  be  the 


centre  of  the  polygon.  Mark  the  altitude  of  this  elementary  tri- 
angle by  a dotted  line  drawn  from  c perpendicular  to  the  base 
a b;  then  multiply  together  the  base  and  altitude  thus  found,  and 
this  product"  by  the  number  of  sides : half  gives  the  area  of  the 
whole  figure. 

Or  otherwise , thus. — Draw  the  straight  line  d e,  equal  to  six 
times,  i.  e.y  as  many  times  a b,  the  base  of  the  elementary  triangle, 
as  there  are  sides  in  the  given  polygon.  Upon  d e describe  an 
isosceles  triangle,  having  the  same  altitude  as  a b c,  the  elementary 
triangle  of  the  given  polygon;  the  triangle  thus  constructed  is 
equal  in  area  to  the  given  hexagon;  consequently,  by  multiplying 
the  base  and  altitude  of  this  triangle  together,  hall  the  product  will 
be  the  area  required.  The  rule  may  be  expressed  in  other  words, 
as  follows  The  area  of  a regular  polygon  is  equal  to  its  perimeter, 
multiplied  by  half  the  radius  of  its  inscribed  circle,  to  which  the 
sides  of  the  polygon  are  tangents. 

Problem  XIX.  — To  describe  the  circumference  of  a circle  through 
three  given  points. 

Let  a,  b,  and  c (Fig.  39)  be  the  given  points  not  in  a straight 
line.  Join  a Band  b c;  bisect  each  of  the  straight  lines  a b and 
n c by  perpendiculars  meeting  in  d *,  then  a,  b and  c are  all  equL 


2d 


PRACTJCAL  CARPENTRY. 


distant  from  d ; therefore  a circle  described  from  d,  with  the  radius 
d a,  d b,  or  d c,  will  pass  through  all  the  three  points  as  required. 

Problem  XX. — To  divide  a given  circle  into  any  7iumber  of  equal 
or  proportional  parts  by  concen  tric  divisions . 

Let  abc  (Fig.  40)  be  the  given  circle,  to  be  divided  into  five 
equal  parts.  Draw  the  radius  a d,  and  divide  it  into  the  same 


number  of  parts  as  those  required  in  the  circle;  and  upon  tne 
radius  thus  divided,  describe  a semicircle:  then  from  each' point  o i 
division  on  a d,  erect  perpendiculars  to  meet  the  semi-circumfer- 
ence in  e,fg,  and  h.  From  d,  the  centre  of  the  given  circle, 
with  radii  extending  to  each  of  the  different  points  of  intersection 
on  the  semicircle,  describe  successive  circles,  and  they  will  divide 
the  given  circle  into  five  parts  of  equal  area  as  required ; the  centre 
part  being  also  a circle,  while  the  other  four  will  be  in  the  form  of 
rings. 

Problem  XXI. — To  divide  a circle  into  three  concentric  parts , 
bearing  to  each  other  the  proportion  of  one,  two,  three,  from  the  centre . 

Draw  the  radius  a d (Fig.  41),  and  divide  it  into  six  equal  parts. 
Upon  the  radius  thus  divided,  describe  a semicircle : from  the  first 
and  third  points  of  division,  draw  perpendiculars  to  meet  the  semi- 
circumference in  e and  f From  d,  the  centre  of  the  given  circle, 
with  radii  extending  to  e and  f describe  circles  which  will  divide 
the  given  circle  into  three  parts,  bearing  to  each  other  the  same 
proportion  as  the  divisions  on  a d,  which  are  as  1,  2 and  3.  In 
like  manner  circles  maybe  divided  in  any  given  ratio  by  concentric 
divisions. 


PRACTICAL  CARPENTRY. 


29 


Problem  XXII. — To  draw  a straight  line  equal  to  any  given  arc 
of  a circle. 

Let  a b (Fig.  42)  be  the  given  arc.  Find  c the  centre  of  the  arc, 
and  complete  the  circle  adb.  Draw  the  diameter  b d,  and  pro- 
duce it  to  e,  until  d e be  equal  to  c d.  Join  a e,  and  extend  it  so 
as  to  meet  a tangent  drawn  from  b in  the  point  f ; then  b f will 
be  nearly  equal  to  the  arc  a b. 

The  following  method  of  finding  the  length  of  an  arc  is  equally 
simple  and  practical,  and  not  less  accurate  than  the  one  just 
given. 

Let  a b (Fig.  43)  be  the  given  arc.  Find  the  centre  c,  and  join 
a b,  b c,  and  c a.  Bisect  the  arc  a b in  d,  and  join  also  c d; 
then  through  the  point  D draw  the  straight  line  e d f,  at  right 


angles  to  c d,  and  meeting  c a and  c b produced  in  e and  f.  Again, 
bisect  the  lines  a e and  b f in  the  points  g and  h.  A straight  line 
g h,  joining  these  points,  will  be  a very  near  approach  to  the  length 
of  the  arc  a b. 

Seeing  that  in  very  small  arcs  the  ratio  of  the  chord  to  the  double 
tangent  or,  which  is  the  same  thing,  that  of  a side  of  the  inscribed 
to  a side  of  the  circumscribing  polygon,  approaches  to  a ratio  of 
equality,  an  arc  may  be  taken  so  small,  that  its  length  shall  differ 
from  either  of  these  sides  by  less  than  any  assignable  quantity; 
therefore,  the  arithmetical  mean  between  the  two  must  differ  from 
the  length  of  the  arc  itself  by  a quantity  less  than  any  that  can  be 
assigned.  Consequently  the  smaller  the  giv^n  arc,  the  more  nearly 
will  the  line  found  by  the  last  method  approximate  to  the  exact 
length  of  the,  arc.  If  the  given  arc  is  above  60  degrees,  or  two- 
thirds  of  a quadrant,  it  ought  to  be  bisected,  and  the  length  of  the 


3° 


PRACTICAL  CARPENTRY. 


semi-arc  thus  found  being  double,  will  give  the  length  of  the  whole 
arc. 

These  problems  arc  very  useful  in  obtaining  the  lengths  of  veneers 
or  other  materials  required  for  bending  round  soffits  of  door  and 
window-heads. 

Problem  XXIII. — To  describe  the  segment  of  a circle  by  means 
of  two  laths , the  chord  and  versed  sine  being  given. 

Take  two  rods,  eb,bf  (Fig.  44),  each  of  which  must  be  at  least 
equal  in  length  to  the  chord  of  the  proposed  segment  ac:  join 
them  together  at  b,  and  expand  them,  so  that  their  edges  shall  pass 
through  the  extremities  of  the  chord,  and  the  angle  where  they  join 
shall  be  on  the  extremity  b of  the  versed  sine  d b,  or  height  of  the 
segment.  Fix  the  rods  in  that  position  by  the  cross  piece  g h, 
then  by  guiding  the  edges  against  pms  in  the  extremities  of  the 
chord  line  a c,  the  curve  abc  will  be  described  by  the  point  b. 

Problem  XXIV. — Having  the  chord  and  versed  sine  of  the  seg- 
ment of  a circle  of  large  radius  given , to  find  any  ?iumber  of  points  in 
the  curve  by  means  of  intersecting  lines. 

Let  a c be  the  chord  and  d b the  versed  sine. 

Through  b (Fig.  45)  draw  e f indefinitely  and  parallel  to  A c; 
join  a b,  and  draw  a e at  right  angles  to  a b.  Draw  also  a o at 


T> 

Fig.  45. 

right  angles  to  a c,  or  divide  a d and  e b into  the  same  number  of 
equal  parts,  and  number  the  divisions  from  a and  e respectively, 
and  join  the  corresponding  numbers  by  the  lines  1 1,  2 2,  3 3. 
Divide  also  a g into  the  same  number  of  equal  parts  as  a d or  e b, 
numbering  the  divisions  from  a upwards,  1,  2,  3,  etc.;  and  from  the 
points  , 2 and  3,  draw  lines  to  b ; and  the  points  of  intersection  ot 
these,  with  the  other  lines  at  h9  h,  /,  will  be  points  in  the  curve  re- 
quired. Same  with  b c. 

Another  Method, — Let  a c (Fig.  46)  be  the  chord  and  d b the 
versed  sine.  Join  a b,  b c,  and  through  b draw  e f parallel  to  a c- 


PRACTICAL  CARPENTRY. 


31 

From  the  centre  b,  with  the  radius  b a or  b c,  describe  the  arcs 
a e,  cf  and  divide  them  into  any  number  of  equal  parts,  as  i*  2, 
3:  from  the  divisions  1,  2,  3,  draw  radii  to  the  centre  B,  and  divide 
each  radius  into  the  same  number  of  equal  parts  as  the  arcs  a b 


Fig.  46. 


and  c F ; and  the  points  g , h , /,  n , 0 , thus  obtained,  are  points  in 

the  required  curve. 

These  methods,  though  not  absolutely  correct,  are  sufficiently 
acqurate  when  the  segment  is  less  than  the  quadrant  of  a circle. 

Problem  XXV. — To  draw  an  ellipse  with  the  trammel . 

The  trammel  is  an  instrument  consisting  of  two  principal  parts, 
the  fixed  part  in  the  form  of  a cross  efgh  (Fig.  47),  and  the 
moveable  piece  or  tracer  klm.  The  fixed  piece,  is  made  of  two 
rectangular  bars  or  pieces  of  wood,  of  equal  thickness,  joined  to- 
gether so  as  to  be  ii.  the  same  plane.  On  one  side  of  the  frame 
so  formed,  a groove  is  made,  forming  a right-angled  cross.  In  the 
groove  two  studs,  k tnd  /,  are  fitted 
to  slide  freely,  and  c trry  attached  to 
them  the  tracer  kin . The  tracer 
is  generally  made  to  slide  ' through  a 
socket  fixed  to  each  stud,  and  pro- 
vided with  a screw  or  wedge,  by 
which  the  distance  a\  >art  of  the  studs 
may  be  regulated.  The  tracer  has 
another  slider  also  adjustable, 
which  carries  a penc  l or  point.  The  instrument  is  used  as  fol- 
lows:— Let  a c be  tl  e major,  and  h b the  minor  axis  of  an  ellipse: 
lay  the  cross  of  the  ti  imrnel  on  these  lines,  so  that  the  centre  lines  of 
it  may  coincide  with  them;  then  adjust  the  sliders  of  the  tracer,  so 
that  the  distance  betveen  k and  m may  be  equal  to  half  the  major 
axis,  and  the  distance  between  / and  m equal  to  half  the  minor 


32 


PRACTICAL  CARPENTRV 


axis ; then  by  moving  the  bar  round,  the  pencil  in  the  slider  will 
describe  the  ellipse. 

Problem  XXVI. — An  ellipse  may  also  be  described  by  means  of  a 
string. 

Let  a b (Fig.  48)  be  the  major  axis,  and  d c the  minor  axis  of 

the  ellipse,  and  f g its  two  foci. 
Take  a string  egf  and  pass  it 
over  the  pins,  and  tie  the  ends 
together,  so  that  when  doubled  it 
may  be  equal  to  the  distance  from 
the  focus  f to  the  end  of  the  axis, 
b;  then  putting  a pencil  in  the 
bight  or  doubling  of  the  string  at 
h and  carrying  it  round,  the  curve 
may  be  traced.  This  is  based  on 
the  well  known  property  of  the 
ellipse,  that  the  sum  of  any  two 
lines  drawn  from  the  foci  to  any 
points  in  the  circumference  is  the  same. 

Problem  XXVII. — The  axes  of  an  ellipse  being  given , to  draw 
the  curve  by  intersections . 

Let  a c (Fig.  49)  be  the  major  axis,  and  d b half  the  minor 
axis.  On  the  major  axis  construct  the  parallelogram  aefc,  and 
make  its  height  equal  to  d b.  Divide  a e and  e b each  into 
the  same  number  of  equal  parts,  and  number  the  divisions  from  a 
and  e respectively;  then  join  ai,  12,  23,  etc.,  and  their  intersec- 
tions will  give  points  through  which  the  curve  may  be  drawn. 

The  points  for  a “ raking  ” or  rampant  ellipse  may  also  be  found 
by  the  intersection  of  lines  as  shown  at  Fig.  50.  Let  a c be  the 
major  and  e b the  minor  axis : draw  a g and  c h each  parallel 
to  b e,  and  equal  to  the  semi-axis  minor.  Divide  a d,  the  semi- 
axis major,  and  the  lines  a g and  c h each  into  the  same  number 
of  equal  parts,  in  r,  2,  3 and  4;  then  from  e,  through  the  divisions 
1,  2,  3 and  4,  on  the  semi-axis  major  a d,  draw  the  lines  e h,  e k,  e /, 
and  e m;  and  from  b,  through  the  divisions  1,  2,  3 and  4 on  the 
line  a g,  draw  the  lines  1,  2,  3 and  4 b;  and  the  intersection  of 


** 


Fig.  48. 


PRACTICAL  CARPENTRY. 


33 


these  with  the  lines  e i,  2,  3 and  4 in  the  points  h kl  m,  will  be 
points  in  the  curve. 

Problem  XXVIII. — To  describe  with  a compass  a figure  resem- 
bling the  ellipse . 

Let  a b (Fig.  51)  be  the  given  axis,  which  divide  into  three  equal 
parts  at  the  points  fg.  From  these  points  as  centres,  with  the  radius 
/a,  describe  circles  which  intersect  each  other,  and  from  the  points 
of  intersection  through  /and  g,  draw  the  diameters  cgEy  c/d. 
From  c as  a centre,  with  the  radius  c d,  describe  the  arc  d e,  which 


Fig.  51.  Fig.  52. 

completes  the  semi-ellipse.  The  other  half  of  the  ellipse  may  be 
completed  in  the  same  manner,  as  shown  by  the  dotted  lines. 

Problem  XXIX. — Another  method  of  describing  a figure  ap- 
proaching the  ellipse  with  a compass . 

The  proportions  of  the  ellipse  may  be  varied  by  altering  the 
ratio  of  the  divisions  of  the  diameter,  as  thus  Divide  the  major 
ax^  of  the  ellipse  a b (Fig.  52),  into  four  equal  parts,  in  the  points 
f g h.  O nfh  construct  an  equilateral  triangle /c  h , and  produce 


34 


PRACTICAL  CARPENTRY. 


the  sides  of  the  triangle  cf  c fi  indefinitely,  as  to  d and  e.  Ther 
from  the  centres  f and  hy  with  the  radius  a f describe  the  circle 
a d gy  b e g\  and  from  the  centre  c,  with  the  radius  c d,  describe 
the  arc  d e to  complete  the  semi-ellipse.  The  other  half  may  be 
completed  in  the  same  manner.  By  this  method  of  construction 
the  minor  axis  is  to  the  major  axis  as  14  to  22. 


**Ai;TlCAL  CARPENTRY. 


3. 


PART  II— ARCHES,  CENTRES,  WINDOW 
AND  DOOR  HEADS. 

N order  that  the  reader  may  be  able  to  “lay  out”  and 
construct  centres  for  arches,  window  and  door  heads,  it 
is  necessary  he  should  have  a clear  conception  of  what 
an  arch  really  is.  For  if  a positive  conclusion  has  not  been  ar- 
rived at,  and  if  the  “ arch  principle  ” is  not  fairly  understood, 
he  cannot  be  expected  to  design  an  arch,  or  to  construct  it  with 
accuracy  or  intelligence,  even  if  designed  by  another.  Let  us  then 
state  once  for  all,  that  every  curved  covering  to  an  aperture  is  not 
necessarily  an  arch.  Thus,  the  stone  which  rests  on  the  piers 
shown  in  Fig.  53  is  not  an  arch,  being  merely  a stone  hewn  out  in 
anarch-like  shape;  but  at  its  top,  the  very  point  (A) 
at  which  strength  is  required,  it  is  the  weakest,  and 
would  fracture  the  moment  any  great  weight  were 
placed  upon  it. 

It  is  not  the  province  of  this  work  to  enter  into  a 
scientific  disquisition  on  the  arch,  but  some  of  its  Fig.  5a 
properties  must  be  known  to  the  mechanic  before  he 
will  be  able  to  construct*  centres  understandingly ; and  the  general 
principles  here  laid  down  will  help  the  workman  materially  to  form 
correct  ideas  concerning  the  work  in  hand.  In  all  cases,  however, 
we  advise  the  student  to  arm  himself  with  a thorough  knowledge  of 
'be  arch  and  the  principles  involved.  Elementary  works  on  the 
subject  can  be  easily  obtained,  and  all  who  would  really  study 
principles,  and  appreciate  the  exquisite  refinement  of  the  examples 
herein  given,  are  strongly  urged  to  read  them. 

The  semi-circular  arch  shown  at  Fig.  54  is  self-explanatory  so  far 
as  the  divisions  are  concerned.  The  under  surface  is  called  the  in- 
trados,  and  the  outer  the  ^trados.  The  supports  are  called  the 


36 


PRACTICAL  CARPENTRY. 


piers  or  abujtments,  though  the  latter  term  is  one  of  more  extensive 
application,  referring  more  generally  to  the  supports  which  bridges 
obtain  from  the  shore  on  each  side  than  to  other  arches.  The  term 
“ piers  ” is,  as  a rule,  supposed  to  imply  supports  which  receive 
vertical  pressure,  whilst  abutments  are  such  as  resist  outward  thrust. 
The  upper  parts  of  the  supports  on  which  an  arch  rests  are  called 
the  imposts.  The  span  of  an  arch  is  the  complete  width  between 
the  points  where  the  intrados  meets  the  imposts  on  either  side;  and 
a line  connecting  these  points  is  called  the  “ springing  ” or  spanning 
line. 

The  separate  wedge-like  stones  composing  an  arch  are  called 
Youssoirs,  the  central  or  uppermost  one  of  which  is  called  the  Key- 
stone ; whilst  those  next  to  the  imposts  are  termed  “ springers.” 


The  highest  point  in  the  intrados  is  called  the  vertex  or  crown , 
and  the  height  of  this  point  above  the  springing  line  is  termed  the 
“rise  ” of  the  arch.  It  will  be  evident  that  in  a semi-circular  arch, 
such  as  Fig.  54,  this  would  be  the  radius  with  which  the  semi-circle 
is  struck.  The  spaces  between  the  vertex  and  the  springing  line 
are  called  the  flanks  or  haunches. 

The  following  are  the  varieties  of  arches  used : — 

The  Semi-circular , as  shown  in  Fig.  54. 

The  Segment  (Fig.  55),  in  which  a portion  only  of  the  circle  is 
used ; the  centre  c is  therefore  not  in  the  springing  line  Sp,  Sp. 

There  are  several  other  kind  of  arches  besides  the  ones  here  de- 
scribed, but  they  are  seldom  made  use  of  by  the  carpenter;  there 
are  the  parabolic,  hyperbolic,  catenarian,  and  cycloidal.  We  have 
given  the  methods  of  describing  the  ellipse,  which,  next  to  the 
circle,  is  the  most  used  in  building. 


$ 

Pig.  55. 


Pig.  54. 


PRACTICAL  CARPENTRY. 


37 


The  Semi-circular  Arch  was  that  principally  used  by  the  Romans, 
who  employed  it  largely  in  their  aqueducts  and  triumphal  arches. 
The  others  are,  however,  mentioned  by  some  writers  as  having  been 


occasionally  employed  by  the  ancients.  During  the  middle  ages 
other  forms  were  gradually  introduced. 

The  Stilted  Arch  is  an  adaptation  of  the  semi-circular  arch,  in 
which  the  springing  line  is  raised  above  the  top  of  the  column,  on  a 
pedestal  not  much  larger  in  diameter  than  the  width  of  the  vous- 
soirs  of  the  arch. 

The  Horse-shoe  Arch . — This  is  almost  restricted  to  the  Arabian 
or  Moorish  style  of  architecture.  In  this  form  of  arch  the  curve  is 
carried  below  the  line  of  centre  or  centres ; for  in  some  cases  the 
arch  is  struck  from  one  centre,  and  in  others  from  two,  as  in  Fig.  56. 

Now  it  must  not  be  supposed  that  the  real  bearing  of  the  arch  is 
at  the  impost  a a ; for  if  this  were  really  so,  it  must  be  seen  that 
any  weight  or  pressure  on  the  crown  of  the  arch  would  cause.it  to 
break  at  b,  but  the  fact  is  simply  that  the  real  bearings  of  the  arch 
are  at  b b,  and  the  prolongation  of  the  arch  beyond  these  points 
is  merely  a matter  of  form  and  has  no  structural  significancy.  The 
Horse-shoe  arch  belongs  especially  to  the  Mohammedan  architec- 
ture, from  its  having  originated  with  that  faith,  and  from  its  having 
been  used  exclusively  by  its  followers. 

Next  in  point  of  time,  but  by  far  the  most  graceful  in  form,  is  the 
pomted  arch,  which  is  essentially  the  mediaeval  (or  middle  age, 
style,  and  is  capable  of  almost  endless  variety.  The  origin  of  this 
form  of  arch  has  been  the  subject  of  much  antiquarian  discussion ; 
but  it  is  certain  that  although  the  pointed  arch  was  first  generally 
used  in  the  architecture  of  the  middle  ages,  recent  discoveries  have 
shown  that  it  was  used  many  centimes  previously  in  Assyria. 


3» 


PRACTICAL  CARPENTRY, 


The  greater  or  less  acuteness  of  the  pointed  arch  depends  on  the 
position  of  the  centres  from  which  the  flanks  are  struck. 

The  Lancet  Arch . — This  arch,  Fig.  57,  is  constructed  by  placing 
the  centres  c c outside  the  span,  but  still  on  the  same  line  with  the 
imposts.  This  form  of  arch  was  first  used  in  the  Gothic,  and  as  a 
rule  indicates  the  style  called  “ Early  English.” 

Equilateral  Arch. — Fig.  58  shows  the  Equilateral  arch,  the  ra- 
dius with  which  tiie  arcs  are  struck  being  equal  to  the  span  of  the 
arch,  and  the  centres  being  the  imposts;  and  thus,  the  crown  and 
the  imposts  being  united,  an  equilateral  triangle  is  formed.  This 
form  was  principally  used  in  the  “ Decorated”  period  of  Gothic 
architecture  from  about  1307  until  about  1390,  at  which  time  the 
Ggee  arch  (Fig.  59)  was  also  occasionally  used. 


At  a later  date,  during  the  existence  of  the  “ Perpendicular  ” 
style  of  Gothic  architecture,  viz.,  from  the  close  of  the  14th  cen- 
tury to  about  1630,  we  find  various  forms  of  arch  introduced,  such 
as  the  Segmental  (Fig.  60),  formed  of  segments  of  two  circles,  the 
centres  of  which  are  placed  below  the  springing;  and  still  later  on 
we  find  the  Tudor \ or  four-centred  arch  (Fig.  61),  in  which  two  of 
the  centres  are  on  the  springing  and  two  below  it.  The  arches  at 
the  later  period  of  this  style  became  flatter  and  flatter,  and  this 
forms  one  of  the  features  of  Debased  Gothic,  when  the  beautiful 
and  graceful  forms  of  that  style  gradually  decayed,  and  for  a time 
were  lost.  Happily,  in  the  present  century  there  has  been  a grad- 
ual and  spirited  revival  of  the  Gothic  style,  and  works  are  now  be- 
ing produced  which  bid  fair  to  rival  in  beauty  of  form  and  in  prin- 
ciples of  construction  the  marvellous  buildings  of  the  middle  ages. 

From  the  examples  given,  the  workman  will  be  able  to  lay  out 
any  of  the  usual  arches  required  in  building. 


Fig.  59. 


Fig.  60. 


Fig.  61. 


PRACTICAL  CARPENTRY. 


39 


There  are  combinations,  however,  of  these  curves  which  the  car- 
penter may  be  called  upon  to  construct,  such  as  the  ones  given 
herewith. 

Fig.  62  is  the  elementary  study  upon  which  the  subsequent  fig- 
ure is  based. 

Having  drawn  the  circle,  describe  on  the  diameter  two  opposite 
semicircles,  meeting  at  the  centre,  a. 

Divine  one  of  these  into  six  equal  parts,  and  set  off  one  of  these 
sixths  from  i to  //. 

Draw  a n , and  divide  it  into  four  equal  parts.  From  the  middle 
point  of  a n draw  a line  passing  through  the  centre  of  the  semi- 


Fig.  62. 


Fig.  63. 


Fig.  64. 


circle,  and  cutting  it  in  From  c set  off  on  this  line  the  lengtn  or 
one  of  the  fourths  of  a //. 

This  point  and  the  two  in  a n will  be  the  centres  for  the  interior 
curves. 

Fig.  63  is  the  further  working  out  of  this  elementary  figure.  It 
is  desirable  that  a larger  circle  should  be  drawn.  Then,  when  the 
figure  has  been  carried  up  to  the  stage  shown  in  the  last,  all  the 
rest  of  the  curves  will  be  drawn  from  the  same  centres. 

Fig.  64  is  the  elementary  form  of  the  tracery  shown  in  Fig.  66. 

We  show  the  method  of  obtaining  these  curves  in  Fig.  67 : At 

any  point,  as  at  a,  draw  a tangent,  and  ag  at  right  angles  to  it. 
From  a,  with  radius  o a,  cut  the  circle  in  b and  c,  and  the  tangent 
in  the  point  f,  using  b as  a centre.  Bisect  the  angle  b at  f,  and 
produce  the  bisecting  line  until  it  cuts  a g in  h.  From  o,  with 
radius  o h,  cut  the  lines  d c and  e b in  1 and  j.  From  h,  i and  j. 


40 


PRACTICAL  CARPENTRY. 


with  radius  H a,  draw  the  three  required  circles,  each  of  which 
should  touch  the  other  two  and  the  outer  circle. 

Returning  now  to  Fig.  64,  having  inscribed  three  equal  circles  in 
a circle,  j ’•*  their  centres,  thus  forming  an  equilateral  triangle. 
From  the  centre  of  the  surrounding  circle  draw  radii  passing  through 
the  angles  of  the  triangle  and  cutting  the  circle  in  points;  as  d and 
two  others.  Draw  e d and  bisect  it  by  eg;  then  thb  centres  for  the 
curves  which  are  in  the  semicircle  will  be  on  the  three  lines  d cf 
€ g and  c e.. 

These  curves,  in  Gothic  architecture,  are  called  “ foliations,1 ” or 
" featherings,1 ” and  the  points  a which  they  meet  are  called  “cusps." 
The  completion  of  this  study  is  shown  at  Fig.  66. 

Fig.  65  shows  the  elementary  construction  f Fig.  63.  Draw  two 
diameters  (Fig.  65)  at  right  angles  to  each  other,  and  join  their  ex- 
tremities, thus  inscribing  a square  in  the  circle.  Bisect  the  quad- 
rants by  two  diameters,  cutting  the  sides  of  the  square  in  the  points, 


as  g;  join  these  points,  and  a second  square  will  be  inscribed 
within  the  first. 

The  middle  points  of  the  sides  of  this  inner  square,  as  bed,  are 
the  centres  of  the  arcs  which  start  from  the  extremities  of  the  diam- 
eters. 

From  b , with  radius  b d , describe  an  arc,  and  from  g,  with  radius 
g cy  describe  another  cutting  the  former  one  in  e.  Then  e is  the 
centre  for  the  arc  i gy  which  will  meet  the  arc  struck  from  by  in  i. 
Of  course,  this  process  is  to  be  carried  on  in  each  of  the  four  lobes. 


PRACTICAL  CARPENTRY. 


41 


Fig.  68  is  the  completed  figure.  The  method  of  drawing  the 
foliation  will  have  been  suggested  by  Fig.  63,  and  is  further  shown 
in  the  present  illustration. 


Fig.  69  shows  the  skeleton  lines  of  Fig.  70.  Divide  the  diameter 
into  four  equal  parts,  and  on  the  middle  two,  as  a common  base, 
construct  the  two  equilateral  triangles  oi  n and  o i m. 

Draw  lines  through  the  middle  points  of  the  sides  of  the  triangles, 
which,  intersecting,  will  complete  a six-pointed  star  in  the  circle, 
the  angles  of  which  will  be  the  centres  for  the  main  lines  of  the 
tracery. 

Fig.  70  is  the  completed  figure. 

The  small  figures,  71  and  72,  will  be  understood  without  further 
instruction  than  is  afforded  by  the  examples. 

Fig.  73  shows  the  construction  of  the  tracery  in  a square  panel. 


71  • Fig.  72.  Pig.  73. 

From  each  of  the  angles  of  the  square  (the  inner  one  in  this 
figure),  with  a radius  equal  to  the  length  of  the  side  of  the  square, 


42 


PRACTICAL  CARPENTRY. 


describe  arcs ; these  intersecting  will  give  a four-sided  curvilinear 
figure  in  the  centre.  Draw  diagonals  in  the  square. 

From  the  point  where  the  diagonal  intersects  the  chrve  b (the 

middle  line  of  the  three  here 
shown)  set  off  on  the  diagonal 
the  length  c b,  viz.,  b m. 

From  q , with  radius  in  qy  de- 
scribe  an  arc  cutting  the  ori- 
ginal arc  in  o. 

Make  in  r equal  to  m <?. 

From  o and  r,  with  radius  ory 
describe  arcs  intersecting  each 
other  in  i : produce  these  until 
they  meet  the  curve  p in  n. 

The  foliation  and  completion 
as  per  Fig.  74  will  now  be  found 
Kiz.  14  simple. 


V'RACl  1CAL  CARPENTRY. 


43 


PART  III— ROOFS. 


ERHAPS  the  best  way  to  “ lay  out”  a common  rafter  is 
by  using  the  “carpenter’s  steel  square,”  and  full  direc- 
tions for  this  purpose  are  given  in  a very  complete  work 
on  “ The  Steel  Square  and  its  Uses,”  published  by  the  Industrial 
Publication  Company,  New  York  City.  As  the  present  work 
is  intended  only  to  show  the  way  the  lengths  and  bevels  can 
be  obtained  by  lines,  we  shall  not  refer  further  to  the  use  of  the 
“ square.” 

There  are  various  kinds  of  roofs,  some  of  them  dependent  for 
their  shapes  on  the  nature  of  the  plan.  The  most  simple  form  of  a 
root  is  the  “leanto”  or  “shed* roof.”  The  roof  most  in  use  is  the 
“saddle-roof,”  which  is  formed  by  two  sets  of  rafters  and  ridge  pole; 


Pig.  75.  Fig.  76.  Pig.  77. 

sometimes  this  is  called  a “ peak-roof.”  Fig.  75  shows  an  elevation, 
and  Fig.  76  a plan  of  this  kind  of  a roof,  while  Figs.  77  and  78 


Pig.  78. 

show  the  plan  and  elevation  of 
two  views  of  a pyrimidal  roof. 


Pig.  79.  Pig  80 

a hip-roof.  Figs.  79  and  80  show 


44 


PRACTICAL  CARPENTRY. 


A few  remarks  on  the  principles  involved  in  roof  construction, 
before  proceeding  further,  will  not  be  out  of  place. 

If  ab,  c b (Fig.  81  a)  be  two  rafters,  placed  on  walls  at  a and  c, 
and  meeting  in  a ridge  b.  Even  by  their  own  weight,  and  much 

more  when 
loaded,  these 
rafters  would 
have  a tenden- 
cy to  spread 
outwards  at  a 
and  c,  and  to 
sink  at  b.  If 
this  tendency 
be  restrained 
by  a tie  estab- 
lished betwixt  a and  c,  and  if  a b,  b c be  perfectly  rigid,  and 
the  tie  a c incapable  of  extension,  b will  become  a fixed  point. 
This,  then,  is  the  ordinary  couple-roof,  in  which  the  tie  a c is  a 
third  piece  of  timber;  and  which  may  be  used  for  spans  of 
limited  extent ; but  when  the  span  is  so  great  that  the  tie  a c 
tends  to  bend  downwards  or  sag,  by  reason  of  its  length,  then  the 
conditions  of  stability  obviously  become  impaired.  Now,  if  from 
the  point  b a string  or  tie  be  let  down  and  attached  to  the  middle 
d,  of  a c,  it  will  evidently  be  impossible  for  a c to  bend  down- 
wards so  long  as  a b,  b c remain  of  the  same  length : d,  therefore, 
like  b,  will  become  a fixed  point,  if  the  tie  b d be  incapable  of  ex- 
tension. But  the  span  may  be  increased,  or  the  size  of  the  rafters 
ab,  cb  be  diminished,  until  the  latter  also  have  a tendency  to  sag; 
and  to  prevent  this,  pieces  de,  df  are  introduced,  extending  from 
the  fixed  point  d to  the  middle  of  each  rafter,  and  establishing 


f and  e as  fixed  points  also,  so  long  as  d E,  d f remain  unaltered 
in  length.  Adopting  the  ordinary  meaning  of  the  verb  “ to  truss,” 
as  expressing  to  tie  up  (and  there  seems  to  be  no  reason  why  we 
should  seek  further  for  the  etymology),  we  truss  or  tie  up  the  point 
d,  and  the  frame  a bc  is  a trussed  frame.  In  like  manner,  f being 
established  as  a fixed  point,  g is  trussed  to  it. 


PRACTICAL  CARPENTRY. 


45 


In  every  trussed  frame  there  must  obviously  be  one  series  of  the 
component  parts  in  a state  of  compression,  and  the  other  in  a state 
of  extension.  The  functions  of  the  former  can  only  be  filled  by 
pieces  which  are  rigid,  while  the  place  of  the  latter  may  be  supplied 
by  strings.  In  the  diagram,  the  pieces  ab,cb  are  compressed,  and 
a c,  d b are  extended  ; yet  in  general  the  tie  d b is  called  a king- 
post,  a term  wnich  conveys  an  altogether  erroneous  idea  of  its 
duties.  Thus  we  see  how  the  two  principal  rafters,  by  their  being 
incapable  of  compression,  and  the  tie-beam  by  its  being  incapa- 
ble of  extension,  serve,  through  the  means  of  the  king-post,  to 
establish  a fixed  point  in  the  centre  of  the  void  spanned  by  the 
roof,  which  again  becomes  the  point  d'appui  of  the  struts,  which  at 
the  same  time  prevent  the  rafters  from  bending,  and  serve  in  the 
establishing  of  other  fixed  points;  and  the  combination  of  these 
pieces  is  called  a king-post  roof. 

It  is  sometimes,  however,  inconvenient  to  have  the  centre  of  the 
space  occupied  by  the  king-post,  especially  where  it  is  necessary  to 
have  apartments  in  the  roof.  In  t jell  a case  recourse  is  had  to  a 
different  manner  of  trussing.  Two  suspending  posts  are  used,  and 
a fourth  element  is  introduced,  namely,  the  straining  beam  a b 
(Fig.  82  b),  extending  between  the  posts.  The  principle  of  trussing 


Fig.  82  Fig  83  c. 


is  the  same.  The  rafters  are  compressed,  the  straining  beam  is 
compressed,  and  the  tie-beam  and  posts,  the  latter  now  called  queen- 
posts , are  in  a state  of  tension. 

In  some  roofs,  for  the  sake  of  effect,  the  tie-beam  does  not 
stretch  across  between  the  feet  of  the  principals,  but  is  interrupted. 
In  point  of  fact,  although  occupying  the  place  of,  it  does  not  fill 


46 


PRACTICAL  CARPENTRY- 


the  office  of  a tie-beam,  but  acts  merely  as  a bracket  attached  to 
the  wall  (Fig.  83  c).  It  is  then  called  a hammer-beam . 

It  is  a general  rule  that  wood  should  be  used  as  struts  and  iron 
as  ties;  and  in  many  modern  trusses  this  rule  has  been  admirably 
exemplified  by  the  combination  of  both  materials  in  the  frames. 

There  is  another  class  of  principals  in  which  tie-beams  are  not 
used.  Such  are  the  curved  principals  of  De  Lorme  and  Emy.  In 
the  system  of  Philibert  de  Lorme,  arcs  formed  of  small  scantlings 
of  timber  are  substituted  for  the  framed  principals;  and  in  that  of 
Colonel  Emy,  laminated  arcs  are  used. 

The  principals  of  roofs  may  therefore,  in  respect  of  their  construc- 
tion, be  divided  broadly  into  two  classes — First,  those  with  tie- 
beams  ; and,  second,  those  without  tie-beams. 

The  first  class,  those  with  tie-beams,  may  be  further  classified  as 
king-post  roofs  and  queen-post  roofs. 

The  second  class  may  be  arranged  as  follows:— 

1 st.  Hammer-beam  roofs. 

2d.  Curved  principal  roofs. 

Having  now  given  such  hints  regarding  the  principles  of  roof 
construction  as  will  enable  the  workman  to  build  any  ordinary  roof 
intelligently,  we  proceed  to  describe  the  methods7  of  construction. 

The  lengths  and  bevels  for  rafters  suitable  for  a roof  similar  to 
that  shown  in  Fig.  75,  are  easily  obtained,  but  the  lines  required 


for  a full  development  of  the  hip-roof  shown  at  Fig.  79,  necessarily 
demands  considerable  skill  and  knowledge  on  the  part  of  the  work- 
man, as  will  be  shown  hereafter. 

In  Fig.  81  we  show  a roof  that  is  at  once  strong  and  cheap  foi 
spans  from  20  to  30  feet,  p p shows  the  wall  plates,  w the  wall 


Fig.  81, 


Fig.  82. 


PRACTICAL  CARPENTRY. 


47 


the  ridge  and  head  of  suspending  rod  ; w and  g show  where  sus- 
pending rods  may  be  placed  if  the  span  exceeds  25  feet. 

Fig.  82  shows  a roof  with  unequal  sides,  a c shows  the  suspend- 
ing rod.  ee  may  be  braces  of  wood  or  rods  of  iron ; b and  n are 
resting  points.  This  is  suitable  for  a span  from  20  to  30  feet. 

Fig.  83  is  suitable  for  a roof  with  a deck,  and  where  the  span  is 
not  more  than  25  feet.  It  is  also  suitable  for  a small  bridge  cross- 
ing a creek,  where  the  span  is  not  more  than  from  16  to  22  feet. 


The  deck  is  shown  at  // ; g /,  s t show  the  suspending  rods ; ab 
show  projections  for  gutters  and  ease-offs. 

Figs.  84  and  85  show  two  schemes  for  timber  roofs.  84  will 
carry  a span  of 


or  a bridge.  If 


used  as  a bridge  it  must  be  braced  herring-bone  style,  as  shown  by 

the  dotted  lines.  The  arch  is 
laminated;  that  is,  formed  of 
thin  pieces  of  timber  bolted 
together.  It  is  suitable  for  a 
span  of  from  zo  to  30  feet. 
Fig.  87  is  a more  preten- 
tious roof  than  any  of  the  foregoing.  It  is  a queen-post  roof  with 
an  iron  king-bolt,  intended  for  a span  of  32  feet. 


Fig.  86. 


48 


PRACTICAL  CARPENTRY. 


A,  is  the  principal  rafter,  5x11  inches. 

b,  straining  beam,  5x11  “ 

c,  queen-post,  5x9  “ 

Struts,  4x5  “ 

c is  the  king-bolt,  and  may  be  .1^  or  1 ^ inches  in  diameter. 
The  common  rafters  may  be  3 x 8 inches,  and  project  over  the  walls 


to  form  the  cornice ; a is  a short  ceiling-joist  of  the  cornice ; b is  an 
ornamental  bracket. 

The  roof  shown  at  Fig.  88  is  one  of  the  best  styles  of  a queen- 
post  roof ; it  is  for  a span  of  40  feet.  The  following  explanations 


*nd  sizes  of  stuff  may  serve  to  aid  the  workman  in  designing  other 
similar  roofs : 

a,  tie-beam  or  chord,  6x12  inches. 

b,  principal  rafter,  6x10  “ 

c,  straining  beam,  6x9  “ 

D,  queen-post,  6x8  “ 


PRACTICAL  CARPENTRY. 


49 


E,  strut,  6x6  inches. 

f,  common  rafter,  6x2|  “ 

a,  wall-plate,  6x9  “ 

by  purlin,  9x12  “ 

Cy  ridge-pole ; d , strap  at  foot  of  common  rafters ; <?,  strap  at  foot 
of  queen-post ; /,  strap  at  head  of  queen-post ; hy  straining  sill. 

Fig.  89  shows  a roof  suitable  for  a span  of  60  feet.  It  has  king- 
post,  queen-post, 
principal,  tie-beam, 
common  rafters,  pur- 
lins and  struts. 

Figs.  90  and  91 
are  combination 
roofs,  and  may  be 
thrown  over  spans 
from  75  to  100  feet. 

One  is  designed  for 
a church  roof,  and 
the  other  for  cover- 
ing a theatre. 


60  It. 


Fig.  89. 


Fig.  92  shows  a scheme  for  a mansard  roof.  This  will  be  found 

to  be  an  excel- 
lent  design 
where  the  span 
is  not  more 
than  32  feet. 

Fig.  93  is  a 
perspect  i ve 
view  of  a king- 
post roof  with 
all  the  various 
pieces  lettered 
for  reference: 
a,  tie-beam;  b, 

wall-plate;  c,  principal  rafter ; d,  kim-post. 

Fig-  94  shows  a perspective  view  of  a queen-post  roof  with  refer* 


TheatFa-rcoi 


Fig.  9CL 


PRACTICAL  CARPENTRY. 


5° 


ence  letters:  a,  tie-beam,  b,  queen-posts,  c,  collar-beam,  d, 
strut,  e,  purlin,  f,  wall-plate,  g,  common  rafters,  h,  ridge-pole. 

A sufficient  number  of  examples  for  timber  roofs  of  the  ordinary 

kind  have  been  shown 
to  enable  the  intelligent 
workman  to  design  and 
construct  any  roof  he 
may  be  called  upon  to 
execute. 

Before  discussing  hip- 
roofs, it  will  be  in  order 
to  give  the  reader  Mr. 
Tredgold’s  rules  for  de- 
termining the  sizes  of 
Fig.  91.  the  various  timbers  for 

the  style  of  roof  exhib- 
ited. It  must  be  borne  in  mind,  however,  that  the  rules  given  are 
empirical,  and  too  general  to  be  relied  on,  except  in  simple  cases. 
It  is  always  best  to  follow  general  usage  in  the  adoption  of  sizes  of 
timber  when  designing  roofs  of  an  unusual  shape  or  character. 


Fig.  93. 


In  estimating  the  pressure  on  a roof,  for  the  purpose  of  appor- 
tioning the  proper  sizes  of  timber  to  be  used,  not  only  the  weight  of 
the  timber  and  the  slates,  or  other  covering,  must  be  taken,  but  also 
the  weight  of  snow  which  in  severe  climates  may  be  on  its  surface, 
and  also  the  force  of  the  wind,  which  we  may  calculate  at  40  lbs. 
per  superficial  foot. 


PRACTICAL  CARPENTRY. 


The  weight  of  the  covering  materials,  and  the  slope  of  *oot 
which  is  usually  given,  are  contained  in  the  following  table:— 


Material. 

Inclination. 

Weight  on 
a Square  FeoL 

Tin  

Kise  1 inch  to  a foot. 

& to  1 1 lbs. 

Copper 

“ 1 “ 

1 ••  3*  » 

Lead 

««  2 ’•  ** 

4 ••  7 •’ 

Zinc 

“ 3 « 

ii  ••  2 ** 

Short  pine  shingles 

(I  5 « M 

n ••  21  ■■ 

Long  cypress  shingles 

<<  g <<  «. 

4 ••  5 “ 

Slate  

I.  0 <1 

5 “ 9 >• 

With  the  aid  of  this  table,  and  taking  into  account  the  pressure 
of  the  wind  and  the 
weight  of  snow,  the 
strength  of  the  different 
parts  may  be  calculated 
from  the  following  em- 
pirical rules,  which  were 
deduced  by  Mr.  Tred- 
gold  from  experience. 

They  are  easy  of  applica- 
tion, and  useful  for  simple 
cases.  Mr.  Tredgold  assumes  66  J lbs.  as  the  weight  on  each  sq.  ft. 

It  is  customary  to  make  the  rafters,  tie-beams,  posts  and  strn;  ail 
of  the  same  thickness. 

IN  A KlNC-POSr  ROOF  OF  PINE  TIMBER. 

To  find  the  dimenzic.i r oj  principal  rafters . 

Rule. — Multiply  the  square  of  the  length  in  feet  by  the  span  in 
feet,  and  divide  the  prciuct  by  the  cube  of  the  thickness  in  incVs \ 
then  multiply  the  quotient  by  0*96  to  obtain  the  depth  in  inches 

Mr.  Tredgold  gives  also  the  following  rule  for  the  rafters,  as  i&ccc 
general  and  reliable  : — 

Multiply  the  square  of  the  span  in  feet  by  the  distance  between 
the  principals  in  feet,  and  divide  the  product  by  60  times  the  rise  to 
feet:  the  quotient  will  be  the  area  of  the  section  of  the  rafter  in  ins. 

If  the  rise  is  one-fourth  of  the  span,  multiply  the  span  by  the  dis- 
tance between  the  principals,  and  divide  by  15  for  the  area  of  section. 


ti  of  ill  us. 


F'V.CTVJa^  carpentry. 


52 

When  the  distance  between  the  principals  is  10  feet,  the  arm  of 
sect’on  is  two-thirds  of  the  span. 

To  find  the  dimensions  of  the  tie-beam , when  it  has  to  support  c 
ceiling  only . ; 

./?///<?. — Divide  the  length  of  the  longest  unsupported  part  by  the 
cube  root  of  the  breadth,  and  the  quotient  multiplied  by  1*4?  will 
give  the  depth  in  inches. 

To  find  the  dimensions  of  the  king-post . 

Rule. — Multiply  the  length  of  the  post  in  feet  by  the  span  in  feet . 
multiply  the  product  by  0*12,  which  will  give  the  area  of  the  sec- 
tion of  the  post  in  inches.  Divide  this  by  the  breadth  for  the  thick- 
ness, or  by  the  thickness  for  the  breadth. 

To  find  the  dimensions  of  struts. 

Rule.— Multiply  the  square  root  of  the  length  supported,  in  feet, 
by  the  length  of  the  strut  in  feet,  and  the  square  root  of  the  product 
multiplied  by  0*8  will  give  the  depth,  which  multiplied  by  0 6 will 
give  the  thickness. 

IN  A QUEEN-POST  ROOF. 

To  find  the  dimensions  of  the  principal  rafters . 

Rule. — Multiply  the  square  of  the  length  in  feet  by  the  span  in 
feet,  and  divide  the  product  by  the  cube  of  the  thickness  in  inches': 
the  quotient  multiplied  by  0.155  g*ve  the  depth. 

7o  find  the  dimensions  of  the  tie-beam. 

Ride. — Divide  the  length  of  the  longest  unsupported  part  by 
the  cube  root  of  the  breadth,  and  the  quotient  multiplied  by  1*47 
will  give  the  depth. 

To  find  the  dimensions  of  the  queen-posts. 

Rule  — — Multiply  the  length  in  feet  of  the  post  by  the  length  in 
feet  of  that  part  of  the  tie-beam  it  supports  : the  product  multiplied 
by  0*27  will  give  the  area  of  the  post  in  inches;  and  the  breadth 
and  thickness  can  be  found  as  in  the  king-post. 

The  dimensions  of  the  struts  are  found  as  before. 

To  find  the  dimensions  of  a straining-beam. 

Rule. — Multiply  the  square  root,  of  the  span  in  feet  by  the  length 
of  the  straming-beam  in  feet,  and  extract  the  square  root  of  the 


PRACTICAL  CARPENTRY. 


product:  multiply  the  result  by  09,  which  wii!  give  the  depth  in 
inches.  The  beam,  to  have  the  greatest  strength,  should  have  i,  5 
depth  to  its  breadth  in  the  ratio  of  10  to  7;  therefore,  to  find  the 
breadth,  multiply  the  depth  by  07. 

To  find  the  dimensions  of pur  lifts. 

Rule. — Multiply  the  cube  of  the  length  of  the  purlin  in  feel  by 
the  distance  the  purlins  are  apart  in  feet,  and  the  fourth  root  of  the 
product  will  give  the  depth  in  inches,  and  the  depth  multiplied  by 
o'  6 will  give  the  thickness. 

To  find  the  dimensions  of  the  common  rafters  when  they  are  placed 
1 2 inches  apart. 

Rule. — Divide  the  length  of  bearing  in  feet  by  the  cube  rc  ot  of 
the  bread  til  in  inches,  and  the  quotient  multiplied  by  072  will  give 
the  depth  in  inches. 

Beams  acting  as  struts  should  noi  be  cut  into  or  mortised  on  one 
side,  so  as  to  cause  lateral  yielding. 

Purlins  should  never  be  framed  into  the  principal  rafters,  but 
should  be  notched.  When  notched,  they  will  ouiry  nearly  twice  as 
much  as  when  framed. 

Purlins  should  be  in  as  long  pieces  as  possible. 

Rafters  laid  horizontally  are  very  good  in  construction,  and  cost 
less  than  purlins  and  common  rafters. 

The  ends  of  tie-beams  should  be  kept  with  a free  space  round 
them,  to  prevent  decay.  It  is  said  that  girders  of  oak  in  the  Chateau 
Roque  d’Ondres,  and  girders  of  fir  in  the  ancient  Benedictine 
monastery  at  Bayonne,  France,  which  had  their  ends  in  the  wall 
wrapped  round  with  plates  of  cork,  were  found  sound,  while  those 
not  so  protected  were  rotten  and  worm-eaten. 

It  is  an  injudicious  practice  to  give  an  excessive  camber  to  the 
tie-beam : it  should  only  be  drawn  up  when  deflected,  as  the  parts 
come  to  their  bearings. 

The  struts  should  always  be  immediately  underneath  that  part  of 

the  rafter  whereon  the  purlin  lies. 

The  diagonal  joints  of  struts  should  be  left  a little  open  at  the 
inner  part,  to  allow  for  the  shrinkage  of  the  heads  and  feet  of  the 
king  and  queen-posts. 


54 


PRACTICAL  CARPENTRY. 


It  should  be  specially  observed  that  all  cranks  or  bends  in  iron 
ties  are  avoided. 

And,  as  an  important  final  maxim — Evety  eomtrmfron  ihould  be 
a little  stronger  than  strong  enough . 

METHODS  OF  DEVELOPING  HIP-ROOFS.* 

The  principles  to  be  determined  in  a hip-roof  are  seven  ; namely: 

ist  The  angle  which  a common  rafter  makes  with  the  level  of 
tiht  top  of  the  building;  that  is,  the  pitch  of  the  roof. 

2nd.  The  angle  which  the  hip-rafter  makes  with  the  level  of  the 
building. 

3d.  The  angles  which  the  hip-rafter  makes  with  the  adjoining 
sides  of  the  roof.  This  is  called  the  backing  of  the  hip. 

4th.  The  height  of  the  roof,  or  the  “ rise,”  as  it  is  called. 

5th.  The  lengths  of  the  common  rafters. 

6th.  The  lengths  of  the  hip-rafters. 

7th.  The  distance  between  the  centre  line  of  the  hip-rafter  and 
the  cents c line  of  the  first  entire  common  rafter. 

The  first,  fourth,  fifth  and  seventh  are  generally  given,  and  from 
these  the  others  may  be  found,  as  will  be  shown  by  the  following 


illustrations:  Let  A BCD  Fig.  95,  be  the  plan  of  a roof.  Draw 

c w parallel;  Co  the  s;des,  a d,  b c,  and  in  the  middle  of  the  distance 
between  them.  T orn  the  points  a b,  c,  d,  with  any  radius,  de- 
scribe the  curves  a b,  & b,  cutting  th  e sides  of  the  plan  at  a , b.  From 

• Taken  from  Steal  Somre  asd  It*  Use*  ** 


PRACTICAL  CARPENTRY. 


55 


these  points  with  any  radius,  bisect  the  four  angles  of  the  plan  at 
r,  r,  r,  r,  and  from  a,  b,  c,  d,  through  the  points,  r,  r,  r,  r,  draw  the 
lines  of  the  hip-rafters,  ag,bg,ch,dh,  cutting  the  ridge-line,  gh, 
in  g and  h,  and  produce  them  indefinitely.  The  cross  lines,  c e, 
dfy  a;e  the  seats  of  the  last  entire  common  rafters.  Through  any 
point  in  the  ridge-line,  i,  draw  e i f at  right  angles  to  g h.  Make 
x K equal  to  the  height  or  rise  of  roof,  and  join  ek,fk;  then  e k 
is  the  length  of  a common  rafter.  Make  go>  h o,  equal  to  i k,  the 
rise  of  the  roof,  and  join  ao,  bo,  c <?,  d o,  for  the  length  of  the  hip- 
rafters.  If  the  triangles,  a og,  bog>  be  turned  round  thtir  seats, 
AOjBG,  until  their  perpendiculars  are  perpendicular  to  the  plane 
of  the  plan,  the  points,  o o,  and  the  lines,  go>  go>  will  coincide, 
and  the  rafters,  a o,  bo>  be  in  their  true  positions. 

If  the  roof  is  irregular,  and  it  is  required  to  keep  the  ridge  level, 
we  proceed  as  shown  in  Fig.  96. 


Bisect  the  angles  of  two  ends  by  the  lines  a by  b by  cg,dg,  in  the 
same  manner  as  in  Fig.  95 ; and  through  g draw  the  lines  G E,  g f, 
parallel  to  the  sides,  cb,da,  respectively  cutting  a by  b by  in  e and 
f;  join  e f ; then  the  triangle,  e g f,  is  a flat,  and  the  remaining 
triangle  and  trapeziums  are  the  inclined  sides.  Join  g by  and  draw 
H 1 perpendicular  to  it ; at  the  points  m and  N,  where  h i cuts  the 


56 


PRACTICAL  CARPENTRY. 


} 

N $ 

/*. 

NT 

VS...- 

yC'"."'X 

7 

— 

lines  G e,  g f,  draw  m k,  n l perpendicular  to  h i,  and  make  them 
equal  to  the  rise ; 
then  draw  h k,  i 
l for  the  lengths 
of  the  common 
rafters.  At£,set 
up  e m perpen- 
dicular tO  B Ej 
make  it  equal  to 
m k or  n l,  and 
join  b m for  the 
lengtn  of  the  hip- 
rafter,  and  pro 
ceed  in  the  same 
manner  to  ob- 
tain A vi,  c m,  97^ 

D vi. 

To  find  the  backing  of  a hip-rafter,  when  the  plan  is  right-angled, 
we  proceed  as  shown  in  Fig.  97.  Let  b b,  b c be  the  common 

rafters,  a d the 
width  of  the  roof, 
and  a b equal  to 
one  half  the 
width.  Bisect  B 
c in  a , and  join 
a a,  d a.  From 
a set  off  a c,  a d 
equal  to  the 
height  of  the  roof 
a b , and  join  a d. 
d c ; then  a d , 0 
c , are  the  hip- 
rafters  To  find 
the  backing* 
from  any  poufi 
h in  a d.  d raw 


PRACTICAL  CARPENTRY. 


57 


w.d  through  g draw 


Piff.  » 


the  perpendicular  h g,  cutting  &Gin  g; 
pendicuiar  to  a a 
the  line  e /,  cut- 
ting a b,  a d in  e 
and  /.  Make  g 
k equal  to  g h, 
and  join  ke,  kf\ 
the  angle  e k / is 
the  angle  of  the 
backing  of  the 
hip-rafter  c. 

Fig.  98  shows 
the  method  of  ob- 
taining the  back- 
ing of  the  hip 
where  the  plan  is 
not  right  angled. 

Bisect  ad  in  a , and  from  a describe  the  semicircle  a^d;  draw 
a b parallel  to  the  sides  a b,  dc,  and  join  a b,  d b,  for  the  seat 
> of  the  hip-rafters. 

From  b set  off 
on  b a,  b d the 
lengths  b d,  b c, 
equal  to  the 
height  of  the  roof 
b c,  and  join  A 
ef  d d ’ for  the 
lengths  of  the 
hip-rafters.  To 
find  the  backing 
of  the  rafters— 
In  a c,  take  any 
point  and  draw 
k A perpendicular 
to  A Through 
h draw fhg\&s- 


5» 


PRACTICAL  CARPENTRY 


pendieular  to  a by  meeting  a b,  a.  p in  / and^-.  Make  k / equal  to 
h ky  and  join  f lyg  /;  then  / /,  i is  the  backing  of  the  hip. 

Fig.  99  shows  how  to  find  the  shoulder  of  purlins: 

First,  where  the  purlin  has  one  cf  its  faces  in  the  plane  of  theioof, 
as  at  E.  From  c as  a centre,  with  any  radius,  describe  the  arc  dg\ 
and  from  the  opposite  extremities  of  the  diameter,  draw  dhy  ^^per- 
pendicular to  B c.  From  e and  f where  the  upper  adjacent  sides  of 
the  purlin  produced  cut  the  curve,  draw  <?•/,//  parallel  to  d/iyg 
tn\  also  draw  c k parallel  to  d k . From  l and  i draw  I in  and  i k 
parallel  to  b c,  and  join  k hy  k in.  Then  c k my  is  the  down  bevel 
of  the  purlin,  and  c k h is  its  side  bevel. 

When  the  purlin  has  two  of  its  sides  parallel  to  the  horizon, 
it  is  worked  out  as  shown  at  f.  ,It  requires  no  further  explana- 
tion. 

When  the  sides  of  the  purlin  make  various  angles  wkh  the  hori- 
zon. Fig.  ioo  shows  the  application  of  the  method  described  in 
Fig.  99  to  these  cases. 

It  sometimes  happens,  particularly  in  railroad  buildings,  that  the 

carpenter  is 
called  upon  to 
pierce  a circu- 
lar or  conical 
roof  with  a 
saddle  roof) 
and  to  accom- 
plish ihh  eco- 
nomically i s 
often  the  re- 
sult of  much 
labor  and  per- 
plexity if  a cor- 
rect method  is 
not  at  hand. 
The  follow- 
ing meth^  shown  in  Fig.  ioi,is  an  excelkm*  one*  and  will  m 
doubt  be  found  useful  in  cases  such  as  mention***! 


PRACTICAL  CARPENTRY. 


59 


Let  d h,  f h be  the  common  rafters  of  the  conical  roof,  and  ic  L» 
S L the  common  rafters  of  the  smaller  roof,  both  of  the  same  pitch. 
On  g h set  up  g e equal  to  m l,  the  height  of  the  lesser  roof,  and 
draw  d parallel  to  d f,  and  from  d draw  cd  perpendicular  to  D F, 
The  triangle  T>dc.>  will  then  by  construction  be  equal  to  the  triangle 
V t m,  and  will  give  tl  e seat  and  the  length  and  pitch  of  the  com- 
men  rafter  of  the  smaller  roof  b.  Divide  the  lines  of  the  seats  in 
b>Ji  figures,  d c>  k m,  into  the  same  number  of  equal  parts;  and 
through  the  points  of  division  in  e,  from  g as  a centre,  describe  the 
curves  c a,  2 g,  if  and  through  those  in  b,  draw  the  lines  3/,  4 g9 
M a,  parallel  to  the  sides  of  the  roof,  and  intersecting,  the  curves  in 
f g a.  ' Through  these  points  trace  the  curves  c fga9  a fg  a}  which 
give  the  lines  of  intersection  of  the  two  roofs.  Then  to  find  the 
valley  rafters,  join  c ay  a a\  and  on  a erect  the  lines  a b9  a h per- 
pendicular to  c a and  a a , and  make  them  respectively  equal  to  M 
L;  then  c by  a b is  the  length  of  the  valley  rafter,  very  nearly. 

Fig.  102  shows  how  a,  curved  hip-rafter  may  be  obtained.  The 


softer  shown  in  this  instance  is  ogee  in  shape,  but  it  makes  no  dif- 
ference what  chape  the  common  rafter  may  be,  the  proper  shape 


6o 


PRACTICAL  CARPENTRY. 


and  length  of  hip  may  be  obtained  by  this  method.  It  will  be  no- 
ticed that  one  side  of  the  example  shown  is  wider  than  the  other; 
this  is  to  show  that  the  rule  will  work  correctly  where  the  sides  are 
unequal  in  width,  as  well  as  where  they  are  equal.  Let  a bcfec 
represent  the  plan  of  the  roof,  fcg  the  profile  of  the  wide  side  of 
the  rafter.  First,  divide  this  rafter,  g c,  into  any  number  of  parts — - 
in  this  case  six.  Transfer  these  points  to  the  mitre  line  e h,  or  what 
is  the  same,  the  line  in  the  plan  representing  the  hip-rafter.  From 
the  points  thus  established  in  e b,  erect  perpendiculars  indefinitely. 
With  the  dividers  take  the  distance  from  the  points  in  the  line  f c, 
measuring  to  the  points  in  the  profile  g c,  and  set  tne  same  off  on 
corresponding  lines,,  measuring  from  e b,  thus  establishing  the 
points  i,  2,  3,  etc.;  then  a line  traced  through  these  points  will  be 
the  required  hip-rafter. 

For  the  common  rafter  on  the  narrow  .side,  continue  the  lines 
from  e b parallel  with  the  lines  of  the  plan  D e and  a d.  Draw  a 
d at  right  angles  to  these  lines.  With,  the  dividers  as  before,  meas- 
uring from  f c to  the  points  in  g c,  set  off  corresponding  dis- 
tances from  a d,  thus  establishing  the  points  shown  between  a and 
h.  A line  traced  through  the  points  thus  obtained  will  be  the  line 
cf  the  rafter  on  the  narrow  side.  This  is  supposed  to  be  the  re- 
turn roof  of  a veranda,  but  is  only  shown  as  an  example,  for  it  is 
not  customary  to  build  verandas  nowadays  with  an  ogee  roof,  but 
with  a rafter  having  a depression  or  cove  in  it.  For  accuracy  it 
would  be  as  well  to  make  nearly  twice  the  number  of.  divisions 
shown  from  i to  6,  as  are  there  represented. 

Fig.  103  shows  a section  of  a Mansard  roof  with  concave  sides, 
and  the  manner  of  framing  the  same  when  it  is  to  be  erected  on  a 
brick  or  stone  building,  p c is  the  wall;  c the  wall-plate;  ab  the 
floor-joist;  hi  is  the  side  rafter;  aie  the  ceiling-joist;  a o the  top 
rafter;  b b d the  bracket  to  nail  cornice  to ; b the  gutter,  and  ri  the 
studding,  which  will  be  required  if  it  is  desirable  to  finish  the  roof- 
story  for  sleeping-rooms. 

The  wall-plate  is  made  of  two  thicknesses  of  two-inch  plank 
nailed  together,  and  lap  jointed  at  the  ends.  The  joists  should 
be  notched  out  to  receive  the  longitudinal  piece  kt  and  the  ends 


PRACTICAL  CARPENTRY. 


61 


of  each  should  be  sawed  off  square  at  or  near  the  dotted  line 
k.  They  should  then  be  put  into  place,  nailed  to  the  wall-plate, 
and  the  piece  h should  be  firmly 
nailed  to  each.  The  lower  end 
of  the  side  rafters  are  cut  out  at 
the  toe  to  rest  on  the  piece  h . 

The  upper  ends  are  also  cut  to 
receive  the  piece  t,  to  which  they 
should  be  firmly  nailed. 

If  it  is  required  to  lath  and 
plaster  on  the  ceiling-joists,  they 
should  be  notched  to  rest  on  the 
piece  i ; but  if  the  room  is  to  re- 
main rough,  it  will  be  as  well  to 
nail  beveled  pieces  on  each  as 
shown  by  the  dotted  line  at  j. 

The  end  of  each  ceiling-joist 
should  be  sawed  in  shape  to  re- 
ceive the  mouVngtf.  with  which 
it  is  usual  to  finish  the  upper 
part  of  the  roof.  The  top  rafters  may  rest  either  on  a longitudinal 
piece  laid  on  the  ceiling-joists,  or  on  the  piece  i — the  latter  being 
the  better  method. 

The  curved  portions  of  the  side  rafters  are  made  separate  from 
the  straight  part,  and  are  most  generally  formed  of  two  thicknesses 
of  inch  stuff,  first  sawed  the  right  shape  and  nailed  together,  and 
then  spiked  to  the  straight  part  of  the  rafter.  When  so  much  of 
the  roof  has  been  put  up,  it  will  be  well  to  mark  on  the  ends  of 
the  floor-joists  the  proper  depth  for  the  gutter.  This  will  be  best 
done  by  holding  a straight-edge  on  the  ends  of  the  joists,  with 
incline  sufficient  to  allow  water  to  run  off,  and  marking  on  each 
joiss  the  depth  it  will  require  to  be  cut  down.  The  vertical  part  of 
the  gutter  is  cut  down  in  a line  with  the  lower  ends  of  the  side 
rafters.  The  cornice  brackets,  which  are  cut  of  a shape  suitable 
for  isceiving  the  different  parts  of  the  cornice,  are  made  of  inch 
stuff,  And  are  nailed  to  the  floor-joists  as  shown  by  the  dotted  Imes 


6<*  PRACTICAL  CARPENTRY. 

artd  nail-marks  at  d k.  The  best  method  to  pursue  in  p^ttijg 
diem  up  is  to  first  nail  one  on  to  the  joist  at  either  extremity  of 
the  roof,  then  stretch  a line  tight  between  the  same  points  on  each, 
and  nail  up  the  intervening  brackets,  with  the  same  points  touching 
the  line.  If  the  line  is  tightly  stretched,  and  proper  care  is  taken 
in  nailing  up  the  brackets,  the  cornice  will  be  perfectly  straight. 

In  Fig.  104  we  have  a section  of  a similar  roof  with  straight 
sides.  The  different  parts  are  lighter  than  those  of  Fig.  103,  and 

the  construction  is  adapted 
for  a balloon  frame  building. 
The  letters  in  Fig-.  104  denote 
the  same  parts  as  the  same 
letters  in  Fig.  "03*  and  ?te 
explanation  of  Fig.  103  will 
answer  for  Fig.  104  so  far  as 
the  same  letters  are  con* 
cerned.  p c is  the  balloon 
frame  studding;  cy  a longi- 
tudinal piece  for  the  floor- 
joists  to  rest  upon.  The 
studs  are  cut  out  at  the  top 
to  receive  the  piece  c , which 
must  be  firmly  nailed  to  each. 
The  floor-joists  are  notched 
to  rest  on  the  piece  ct  and 
will  thus  prevent  the  flams 
from  spreading. 

Smce  there  is  no  curve  on  the  rafter,  the  face  of  it  comes  flush 
with  the  inside  of  the  gutter.  Hence  the  side  rafters  ? re  cut  out 
at  the  heel  to  rest  on  the  piece  //,  instead  of  the  toe,  as  in  Fig.  103. 
The  piece  h is  beveled  in  order  that  the  thrust  on  the  side  rafters 
shall  not  throw  the  lower  ends  out.  The  inside  of  the  gutter  is  also 
made  inclining  so  as  to  give  as  much  substance  as  possible  between 
the  gutter  and  the  piece  h.  The  remaining  parts  are  the  sane  as 
those  Fig.  103,  and  the  same  description  of  those  parts  will 
answer  foi  both  cuts. 


PRACTICAL  CARPENTRY. 


63 


Fig.  505  shows  how  to  find  the  angle-rafter  and  angle-cornice 
bracket,  when  the  section  as  above  described  has  been  drawn. 
Let  abc  represent  the  given  section  on  the  draughting-board  or 
floor,  in  which  the  same  letters  denote  similar  parts  in  Figs.  103 
and  104.  Draw  the  line  A o at  an  angle  of  450  with  a f.  Then 
from  any  points  c, /,  oy  etc.,  of  the  section  as  shown,  draw  lines 
perpendicular  to  a f,  and  intersecting  a o.  In  order  to  transfer 
the  distances  a e,  a etc.,  on  a 0 to  a h,  it  is  most  convenient; 


in  our  small  illustration,  to  describe  arcs  with  a as  a centre;  but  in 
practice,  since  the  distance  a o will  be  several  feet,  it  will  be  best 
io  lay  a straight-edge  along  the  line  a oy  and  mark  the  points  a,  e, 
P , etc.,  on  it ; then  change  the  position  of  the  straight-edge,  and 
lay  it  along  a h — the  point  before  on  a being  made  to  coincide 
\vith  it  again,  and  transfer  the  marks  to  the  floor  or  board  on  the 
line  a h at  e',/",  etc.  When  this  has  been  done,  draw  lines  from 


64 


PRACTICAL  CARPENTRY. 


these  marks  and  perpendicular  to  ah.  Now  draw  lines  from  the 
points  Cypy  o,  etc,,  on  the  section  a b c,  but  parallel  to  f h,  and  in- 
tersecting the  lines  which  are  perpendicular  to  a h.  Note  the  in- 
tersection of  any  two  of  these  lines  which  were  produced  from  the 
same  point  of  the  section,  and  this  intersection  will  be  the  similar 
point  of  the  angle-rafter.  Perhaps  the  subject  will  be  better  under- 
stood if  we  follow  the  details  of  finding  a single  point  of  the  angle- 
r;after ; such,  for  instance,  as  that  corresponding  to  the  point  p oi 
the  given  section.  From  p draw  pp'  perpendicular  to  a f,  and  in- 
tersecting a o at  p '.  Make  the  distance  a pf  on  ah  equal  to 
a p'  on  a Oy  either  by  describing  an  arc  with  a as  a centre  and 
a p'  as  radius,  or  by  transferring  the  point  p to  p"  on  a straight- 
edge, as  before  stated.  From  p"  draw  p"  p"  perpendicular  to  a h. 
Then  from  p on  the  section  draw  a line  pp”  parallel  to  f h,  until  it 
intersects  the  line  f p"  in  the  point  This  point  p”  will  be  the 
point  of  the  angle-rafter  corresponding  to  the  point  p of  the  section. 
After  finding  all  the  points  in  a similar  manner,  they  must  be  joined 
by  the  requisite  curved  line,  and  a pattern-rafter  cut  to  fit  It  will 
be  apparent  from  inspection  that  the  angle-bracket  is  found  in  the 
same  manner. 


PRACTICAL  CARPENTRY. 


6S 


PART  IT- COVERING  OF  ROOFS. 


N slating  or  shingling  a roof,  great  care  should  be  taken 
at  the  hips,  ridges  and  valleys.  Where  the  roof  is 
shingled,  two  or  three  courses  should  be  left  off  at  the 
ridge  until  the  two  sides. are  brought  up,  then  the  courses  left  oft 
should  be  laid  on  together,  and  in  such  a manner  as  to  have  them 
“ lap  ” over  each  other  alternately.  This  can  be  easily  done  if  the 
workman  uses  a little  judgment  in  the  matter;  and  a roof  shingled 
in  this  manner  will  be  perfectly  rain-tight,  without  the  ridge-boards 
or  cresting.  In  valleys,  the  tin  laid  in  should  be  sufficiently  wide 
to  run  up  the  adjacent  sides  far  enough  to  prevent  “ back-flow  ” 
from  running  over  it.  Ample  space  should  also  be  left  in  the 
gutter  to  permit  the  water  to  flow  off  freely.  There  is  a general 
tendency  to  make  these  waterways  too  narrow,  which  is  frequently 
the  cause  of  the  water  backing  up  under  the  shingles,  causing 
leakage  and  premature  decay  of  roof. 

There  are  several  methods  of  shingling  over  a hip-ridge ; we 


prefer,  however,  the  old  and  well-tried  method  of  shingling  with  the 
edges  of  the  shingles  so  cut  that  the  grain  of  the  wood  runs  parallel 
with  the  line  of  hip,  as  shown  in  Fig.  xo6.  Here  it  will  be  seen 


PRACTICAL  CARPENTRY. 


6G 

that  the  shingles  next  to  those  on  the  hip  have  the  grain  running 
up  and  down  at  right  angles  with  the  eave.  On  Fig.  107  we  show 
a front  view  of  the  same  hip,  which  will  give  a better  idea  of  what 
is  meant  by  having  the  grain  parallel  with  the  line  of  hip.  abed 
show  the  cut  or  hip  shingles,  and  n n n n the  common  shingles. 

The  proper  way  to  put  in  these  shingles  is  to  let  the  ends  run 
over  alternately  and  then  dress  them  to  the  bevel  of  the  opposite 
side  of  the  roof ; this  is  shown  better  at  o b d>  where  the  edges  of  the 
shingles  are  shown  that  are  laid  on  the  other  side  of  the  roof. 
The  edges  of  a and  c show  on  the  other  side  of  the  roof,  and  are  not 
seen  in  the  drawing. 

To  cover  a circular  dome  with  horizontal  boarding,  proceed  as 
follows : 

Let  abc  (Fig.  108)  be  a vertical  section  through  the  axis  of  a 


circular  dome,  and  let  it  be  required  to  cover  this  dome  horizon- 
tally. Bisect  the  base  in  the  point  d,  and  draw  p b e perpendicular 
to  a c,  cutting  the  circumference  in  b.  Now  divide  the  arc  b c 


PRACTICAL  CARPENTRY, 


6* 


too  equal  parts,  so  that  each  part  will  be  rather  less  than  the 
width  of  a board ; and  join  the  points  of  division  by  straight  lines, 
which  will  form  an  inscribed  polygon  of  so  many  sides ; and 
through  these  points  draw  lines  parallel  to  the  base  a c,  meeting 
the  opposite  sides  of  the  circumference.  The  trapezoids  formed  by 
the  sides  of  the  polygon  and  the  horizontal  lines,  may  then  be  re- 
garded as  the  sections  of  so  many  frustrums  of  cones;  whence 
results  the -following  mode  of  procedure;  produce,  until  they  meet 
the  line  d e,  the 
lines  nf9  fg9  etc., 
forming  the  sides 
of  the  polygon. 

Then  to  describe 
a board  which  cor- 
responds to  the 
surface  of  one  of 
the  zones,  as  f g9 
of  which  the  trape- 
zoid is  a section — 
from  the  point  h9 
where  the  line  fg 
produced  meets  d 
e,  with  the  radii 
h f9  h g9  describe 
two  arcs,  and  cut 
off  the  end  of  the 
board  k on  the 
line  of  a radius  k 
k.  The  other 
boards  are  de- 
scribed in  the 
same  manner. 

To  describe  the 


Fig.  109. 


(Servering  of  an  ellipsoidal  dome  with  boards  of  equal  width . 

Let  a bcd  (No.  i,  Fig.  109)  be  the  plan  of  the  dome,  abc 
{No.  2)  the  section  on  its  major  axis,  and  l m n (No.  3)  the  section 


68 


PRACTICAL  CARPENTRY. 


on  its  minor  axis.  Draw  the  circumscribing  parallelogram  of  the 
ellipse  f g h k (No.  i),  and  its  diagonals  f h g k.  In  No.  2 
divide  the  circumference  into  equal  parts  1234,  representing  the 
number  of  covering*  boards,  and  through  the  points  of  division  1 8, 
2 7,  etc.,  draw  lines  parallel  to  a c Through  the  points  of  division 
draw  1 /,  2 /,  3 x,  etc.,  perpendicular  to  a c,  cutting  the  diagonals 
of  the  circumscribing  parallelogram  of  the  ellipse  (No.  1),  and 
meeting  its  major  axis  in  p t xy  etc.  Complete  the  parallelograms, 
and  their  inscribed  ellipses  corresponding  to  the  lines  of  the  cover- 
ing, as  in  the  figure.  Produce  the  sides  of  the  parallelograms  to 
intersect  the  circumference  of  the  section  on  the  transverse  axis  of 
the  ellipse  in  1 2 3 4,  and  lines  drawn  through  these,  parallel  to  l 
n,  will  give  the  representation  of  the  covering  boards  in  that  sec- 
tion. To  find  the  development  of  the  covering,  produce  the  axis 
d B,  in  No.  2,  indefinitely.  Join  by  a straight  line  the  divisions  1 
2 in  the  circumference,  and  produce  the  line  to  meet  the  axis  pro- 
duced; and  12  e kg  will  be  the  axis  major  of  the  concentric  ellipses 
of  the  covering  i f g,  2 h k . Join  also  the  corresponding  divisions 
in  the  circumference  of  the  section  on  the  minor  axis,  and  produce 
the  line  to  meet  the  axis  produced  ; and  the  length  of  this  line  will 
be  the  axis  minor  of  the  ellipses  of  the  covering  boards. 

Before  leaving  the  subject  of  roofs,  it  may  be  a?  well  to  remark 
that  the  framing  of  valley  roofs  is  so  very  much  like  that  of  hip- 
roofs, that  it  was  not  necessary  to  make  special  engravings  for  the 
purposes  of  showing  how  a valley-roof  is  constructed  or  “ laid  out.” 
The  cuts,  bevels,  lengths  and  positions  of  rafters  and  jacks  may  be 
easily  found  if  the  same  principles  that  goierr  Vip-roofs  cxefcl- 
lowed,  as  a valley  rafter  is  simply  a hip  reversed. 


PRACTICAL  CARPENTRY. 


69 


PART  T.— MITERING  MOULDINGS, 

NE  of  the  most  troublesome  things  the  carpenter  meets 
with  is  the  cutting  of  a spring  moulding  when  the  hori- 
zontal portion  has  to  mitre  with  a gable  orraking  mould- 
ing. Undoubtedly  the  best  way  to  make  good  work  of  these 
mouldirgs  is  to  use  a mitre-box.  To  do  this  make  the  down  cuts 
B,  b (Fig.  no)  the  same  pitch  as  the  plumb  cut  oh  the  rake.  The 
over  cuts  o,  o,  o,  o should  be  obtained  as  follows : Suppose  the 

roof  to  be  a quarter  pitch— though  the  rule  works  for  any  pitch 


wnen  followed  as  here  laid  down — we  set  up  one  foot  of  the  rafter, 
as  at  Fig.  in,  raising  it  up  6 inches,  which  gives  it  an  inclination 

of  quarter  pitch ; then  the  diagonal 
will  be  nearly  13*4  inches.  Now 
draw  a right-angled  triangle  whose 
two  sides  forming  the  right  angle, 
measure  respectively  12  and  13J4 
inches,  as  shown  in  Fig.  112. 

"The  lines  a and  b show  the  top  of  the  mitre-box  with  the  lines 


70 


PRACTICAL  CARPENTRY, 


marked  on.  The  side  marked  13^ 
inches  is  the  side  to  mark  from ; this 
must  be  borne  in  mind,  and  it  must 
be  remembered  that  this  bevel  must 
be  used  for  both  cuts,  the  12  inch 
side  not  being  used  at  all. 

Another  excellent  method  for  obtaining  the  section  of  a raking 
mould  that  will  intersect  a given  horizontal  moulding,  is  given 
below,  also  the  manner  of  finding  the  cuts  for  a mitre-box  for  same. 
The  principles  on  which  the  method  is  based  being,  first,  that 
similar  points  on  the  rake  and  horizontal  parts  of  a cornice  are 
equally  distant  from  vertical  phnes  represented  by  the  walls  ot  a 
building;  and,  second,  that  such  similar  points  are  equally  distant 
from  the  plane  of  the  roof.  Representing  the  wall  faces  of  a build- 
ing bv  the  line  d b (Fig.  113),  and  a section  of  the  horizontal  cor- 
nice by  d b ab  cde  f—  b a b c being  the  angle  of  the  roof  pitch— 
and  following  the  idea  given  in  Figure  105,  draw  lines  aa',c  f\ 
parallel  to  d b and  intersecting  the  line  k a\  which  is  drawn  at 
right  angles  to  d b through  the  point  b ; then,  with  b as  a centre, 
describe  the  arcs  a'  c'  /',  f etc.,  intersecting  the  same  line  k 
a'  on  the  opposite  side  of  d b ; after  which  extend  lines  from  the 
points  d b k , parallel  to  d b.  This  gives  the  point  k at  the  same 
distance  from  d B-as  the  points  a and  a\  and  the  line  lb  at  the 
same  distance  as  c d.  The  rest  of  the  same  group  of  parallel  lines 
are  found  to  be  similarly  situated  with  respect  to  a b. 

From  Descriptive  Geometry  we  have  the  principle,  that  if  we 
have  given  two  intersecting  lines  contained  in  a plane,  we  know 
the  position  of  that  plane ; hence  we  may  represent  the  plane  of  a 
roof  by  the  line  b a and  b k (Figs.  113  and  114)1  anc^  s*nce 
be  most  convenient  to  measure  the  distances  required  in  a direction 
perpendicular  to  that  plane,  in  following  out  the  principle  draw  lines 
from  the  points  c e ft  etc.,  parallel  to  b a and  intersecting  the  line 
b g,  which  is  made  perpendicular  to  b a.  This  gives  us  on  b^  the 
perpendicular  distance  of  the  points  c e f etc.,  from  the  line  b <7. 
From  the  intersections  of  these  lines  with  b g , and  with  b as  a 
centre,  describe  arcs  intersecting  the  line  d b at  i hr g\  etc.;  from 


Fig.  112. 


PRACTICAL  CARPENTRY. 


V 


these  intersections  with  d b draw  lines  i'  /,  h'  p}  g'  r , etc,  parafiel  to 
b ky  until  they  in- 
tersect  the  first 
group  of  lines 
drawn  perpendicu- 
lar to  b ky  and  the 
intersection  of  each 
set  of  two  lines 
drawn  from  the 
same  pomt  on  the 
horizontal  section 
will  give  the  simi- 
lar point  of  the  rake 
section.  Takin  g 
the  point  /,  for  ex- 
ample. we  have,  as 
before  proved,  / at 
the  same  distance 
from  d e as  c,  and 
i being  at  the  same 
distance  from  b a as  a d i being  equal  to  b i\  and  b i = 1 t,  i 
( is  equal  to  b : and  consequently,  / is  the  same  distance 

from  b k as  c is  from 
b ay  which  is  in  ac- 
cordance with  princi- 
ple already  shown. 
The  intersection  of 
each  set  of  lines  be- 
ing found  and  marked 
by  a point,  the  con- 
tour of  the  moulding 
may  be  sketched  in. 
Fig.  H4.  and  the  rake  mould- 

ing, of  which  the  scc- 

$nn  is  thus  found,  will  intersect  the  given  horizontal  moulding, 
4 proper  care  has  been  taken  in  executing  the  diagram. 


P^LllCAL  CAIU&NTMY* 


i* 


t:  * 


Fig.  115. 

Fig.  1 15  shows  ho^  to  find  die  mitre  cut  for  the  rake  moulding, 
the  cut  for  the  horizonta;  cne  being  the  same  as  for  any  ordinary 


9tg  in 


PRACTICAL  CARPENTRY. 


7* 


moulding.  Take  an  ordinary  plain  mitre-box,  n j l,  and  draw  the 
line  a b,  making  the  angle  abj  equal  to  the  pitch  angle  of  the 
roof.  Draw  b d perpendicular  to  a b,  and  make  it  equal  to  the  width 
of  the  box  k j ; make  d e parallel  to  a b,  and  extend  lines  from  b 
and  p square  across  the  box  to  k and  c ; join  b c and  e k.  abc 
will  be  the  mitre  cut  for  two  of  the  rake  angles;  hek  will  be  the 

cut  for  the  other  two  angles,  the  angle  hen  being  equal  to  the 

^ngle  abj.  In  mitering,  both  horizontal  and  rake  moulding,  that 
part  of  the  moulding  which  ‘is  vertical  when  in  its  place  on  the 
cornice,  must  be  placed  against  the  side  of  the  box. 

Lines  for  the  cuts  m a mitre  box,  for  joining  spring  mouldings 
may  be  obtained  as  follows:  If  we  make  b Fig.  216,  the  moulding 

showing  the  spring  or  lean  of  the  member,  and  d e the  mitre  re- 
quired,  then  proceed  as  follows : With  a as  a centre,  and  the  radius 
a g,  describe  the  semicircle  fhg  c;  then  drop  perpendiculars 
Srcm  the  line  f c,  at  the  points  f,  a,  h,  g and  c,  cutting  the  mitre 
line  as  shown  on  the  line  1 d.  Draw  1 e parallel  to  f c,  then  from 

2 draw  i s,  which  will  be  the  bevel  for  the  side  of  the  box,  and  the 

bevel  o r will  be  the  line  across  the  top  of  the  box.  The  mitre 
line,  as  shown  here,  is  for  an  octagon,  but  the  system  is  applicable 
to  any  figure  from  a triangle  or  rectangle  to  & polygon  with  any 
number  of  sides. 


PRACTICAL  carpentry; 


PART  VI  -SASHES  AND  SKYLIGHTS. 


N the  skylight,  Fig.  117,  of  which  No.  i is  the  plan,  and 
No,  2 the  elevation,  it  is  required  to  find  the  length  and 
backing  of  the  hip. 


Let  a b be  the  seat  of  the  hip ; erect  the  perpendicular  a c,  and 


Pig.  n? . 


PRACTICAL  CARPENTRY. 


75 


make  it  equal  to  the  vertical  height  of  the  skvlighi,  and  draw  b c, 
which  is  the  line  of  the  underside  of  the  hip.  The  dotted  hne^  k 


Fig.  us. 

To  find  the  backing,  from  any  point  in  b c,  as^:  draw  perpen- 
dicular to  BC,a  line  g f meeting  a b in  f,  and  through  f draw  a 


i7iT>  e/tm  e>  /it 


PRACTICAL  CARPENTRY. 


16 


line  at  right  angles  to 
a b,  meeting  the  sides 
of  the  skylight  in  d 
and  e.  Then  from  f 
as  a centre,,  and  .with 
F£-as  radius,  cut  the 
line  a b in  //,  and  join 
d //,  e //.  The  angle 
d^e  is  the  backing 
of  the  hip,  and  th* 
bevel  d h e will  give 
the^  angle  of  backing 
when  applied  to  the 
perpendicular  side  of 
the  hip  bar. 

In  Fig.  1 1 8,  in 
which  No.  i is  the 
plan,  and  No.  2 the 
elevation  of  a skylight 
with  curved  bars,  to 
find  the  hip:  let  a b be 
the  seat  of  the  centre 
bar,  and  d e the  seat 
of  the  hip.  Through 
any  divisions  1234c 
of  the  rib,  over  a b 
draw  lines  at  right 
angles  to  a b and  pro- 
duce them  to  meet  e d 
in porsv.  From  these 
points  draw  lines  per- 
pendicular to  e D,  and 
set  up  on  them  the 
corresponding  heights 
from  a b t m n 1 0 t, 
m 2 in \ p a,  eitv 


1P21ACT3CAL  CARPENTRY. 


u 


Fig.  £19  shows  several  ribs  suitable  for  skylights.  They  aie  de- 
signedly made  complicated  so  as  to  exemplify  the  manner  of  getting 
the  shapes  of  the  mouldings.  No.  1 shows  the  section  of  a rib; 
these  ribs  may  be  moulded  as  shown,  or  they  may  be  chamfered 
from  the  glass  line  down  to  the  point  a.  No.  2 shows  a hip  01 
angle  rib;  the  backing,  qds,  is  obtained  as  shown  in  Figs.  1 17 
and  r s3.  No.  4 is  Another  hip  made  of  larger  section  than  No.  2. 
No&  ? and  5 show  sections  of  bars  that  may  be  used  in  connection 
with  the  ribs  where  required.  No.  3 is  drawn  on  an  angle  and  let* 
tered  for  reference,  so  as  to  show  the  workman  how  such  bars, 
mouldings  or  other  work  can  be  manipulated  when  the  necessity 
for  their  use  arises. 

Figs.  120,  12 1 and  122  shew  how  an  angle,  bar  for  ordinary 


Si*.  120. 


PRACTICAL  CARPENTRY. 


78 

sashes  may  be  obtained.  Fig.  121  exhibits  a section  of  the  regular 
bar,  which  may  be  any  shape.  The  lines  abed  are  drawn  from 
fixed  points  of  the  moulding.  These  lines  are  continued ; they  cut 
the  lines  o,  o in  Figs.  120  and  122.  Make  the  distances  on  the 
Hues  abed , etc,,  in  Figs.  120  and  122,  the  same  as  in  Fig.  121, 
»*om  the  line  o.  , The  points  of  juncture  of  these  lines  with  the 
Junes  parallel  with  the  central  sectional  lines  o o,  will  be  the  points 
through  which  to  describe  the  angle  bar. 

Fig.  120  shows  a bar  set  on  an  angle  of  450,  or,  as  workmen 
term  it.  “ it  is  a mitre  bar.”  Fig.  122  is  set  on  a more  oblique 
angle.  The  rules  given  in  the  foregoing  will  apply  to  any  angle 


A very  ready  way  to  find  the  shape  of  an  angle  bar  is  to  take  a 
piece  of  the  straight  bar  and  stand  it  on  edge  in  the  mitre  box,  and 
saw  off  a thin  section  of  the  bar  to  the  same  angle  as  the  bar  re- 
quired; then  the  outlines  of  this  thin  section  will  be  very  nearly  the 
shape  wanted.  Some  workmen  adopt  this  method  altogether  of 
finding  the  section  of  their  angle  bars,  but  we  do  not  recommend  it 
£S  it  is  faulty  in  more  than  one  respect,  and  is  unscientific. 


PRACTICAL  CARPENTRY. 


79 


PART  VIL— MOULDINGS. 


NGLE  brackets  for  coves  or  any  other  mouldings  may  be 
laid  off  by  proceeding  as  follows  (Fig.  123) : First,  when 
it  is  a mitre  bracket  in  an  interior  angle,  the  angle  being 
450,  divide  the  curve  c b into  any  number  of  equal  parts  12345, 


h/id  draw  through  the  divisions  the  lines  t dt  3/  4<f,  ps? 


PRACTICAL  CARPENTRY. 


to 

pendicular  to  a b,  and  cutting  it  in  def gc\  and  produce  them  to 
meet  the  line  de,  representing  the  centre  of  the  seat  of  the  angle 
bracket;  and  from  the  points  of  intersection,  kiklcy  draw  lines 
h i,  12,  k 3, 1 4,  at  right  angles  to  d e,  and  make  them  equal — h i 
to  d i,  i 2 to  e 2,  etc.;  and  through  f 12345  draw  the  curve  of 
the  edge  of  the  bracket.  The  dotted  lines  on  each  side  of  de  on 
the  plan  show  the  thickness  of  the  bracket,  and  the  dotted  lines 
u r,  v s,  w /,  show  the  manner  of  finding  the  bevel  of  the  face.  In 
the  same  figure  is  shown  a method  for  finding  the  bracket  for  an 
obtuse  exterior  angle.  Let  o t k be  the  exterior  angle ; bisect  it  by 


the  line  t g , which  will  represent  the  seat  of  thecentre  of  the  bracket. 
The  lines  t h,  tn  i,  n 2,  o 3,  p 4,  c 5,  are  draw^i perpendicular  to  t g, 


PRACTICAL  CAkPEHl  RV 


6 1 


nnd  their  lengths  are  found  as  in  the  former  case.  The  bracket  foi 
c cute  angle  may  also  be  found  by  a like  process. 

To  find  the  angle  bracket  at  the  meeting  of  a concave  curved 
wall  with  a straight  wall  we  proceed  as  follows  s Lef  adbr  (Fig. 
S24)  be  the  plan  of  the  bracketing  on  the  straight  wall,  and  d m,  k 
O the  plan  on  the  circular  wall ; c a b the  elevation  on  the  straight 
wall,  and  g m h on  the  circular  wall.  Divide  the  curves  cb,gh 
hvo  the  same  number  of  equal  parts;  through  the  divisions  of  c B 
draw  the  lines  c d,  i d h,  2 e i,  etc.,  perpendicular  to  a b,  and 
through  those  of  g h draw  the  parallel  lines,  part  straight  and  part 
curved,  1 m li , 2 n i,  3 0 k,  etc.  Then  through  the  intersections  h i 
k l of  the  straight  and  curved  lines,  draw  the  curve  d e,  which  will 
give  the  line  from  which  to  measure  the  ordinates  h i,  i 2,  £3,  etc. 
Angle  brackets  for  any  corner  may  be  found  by  this  process  if  a 
little  judgment  is  displayed  in  applying  the  rule. 

Fig.  125  shows  the  manner  of  finding  the  proportions  of  a ima!! 
moulding  which  is  required  to  mitre  with  a larger  one,  or  vice  vers<s. 


Fig.  125. 


Let  a b be  the  length  of  the  larger  moulding,  and  a d the  length  of 
the  smaller  one;  construct  with  these  dimensions  the  parallelogram 
a d c b,  and  draw  its  diagonal  a c ; draw  parallel  to  b c lines  as,  b 
etc.,  etc.,  meeting  the  diagonal  in  s t , etc.,  and  from  these  points 
draw  parallels  to  a b,  meeting  ad  in  n op  r.  produce  them  to  i k 
tm,  etc.,  and  make  n i equal  to  e a,  0 k to  f b,  etc,  and  thus  coo 
pJete  the  contour  of  the  moulding  on  d a,  the  lengths  of  which  are 
diminished  in  the  ratio  of  ad  to  ab,  but  its  projections  remain  the 


PRACTICAL  CARPENTRY. 


s&me  as  those  of  the  larger  moulding.  The  operation  may  be  &G 
versed,  and  the  larger  produced  from  the  smaller  moulding. 

Fig.  126  shows  the  manner  of  enlarging  or  diminishing  a single 
moulding.  Let  ab  be  a moulding  which  it  is  required  to  reduce 
to  a d.  Make  the  sides  a b,  d c,  and  ad,  b c of  the  parallelograms 
respectively  equal  to  the  larger  and  smaller  moulding,  and  drav  the 
diagonal  a c,  produce  da  to  e,  and  make  eaf  equal  to  at  A,  nd 


draw  a f.  The  manner  of  obtaining  the  lengths  and  projective 
with  these  data  is  so  obvious  that  further  description  is  unnecessary. 

To  get  the  contour  or  outline  for  a ra&ng  moulding,  proceed  as 
shown  in  Fig.  127.  The  horizontal  noulding  is  divided  into  any 
number  of  parts,  equal  ( or  unequal,  a3  shown  at  ab  ede  f g.  The 
line  a c shows  the  rake  or  inclination.  Draw  lines  parallel  with  a 
c,  from  a to  d,  b to  v , etc.  Drop  a line  a b,  perpendicular  to  a c, 
at  any  convenient  point  on  the  rake,  make  the  distance  a c equai  to 
ho;  then  drop  the  lines  / qr  s t,  and  where  these  lines  cut  the  linsa 
a b c 'd  e fg,  these  points  of  contact  will  be  in  the  curve  line  of  the 
raking  moulding,  as  at  d vivxyz.  From  these  points  trace  the 
curve,  which  will  be  the  proper  shape  for  the  moulding.  The  divi- 
sions and  lines  shown  at  ghef  gives  the  proper  shape  for  die 
moulding  at  the  top  return.  It  requires  no  further  explanation. 

Fig.  128  shows  another  application  of  the  foregoing  rule.  This 
will  apply  to  raking  string-boards,  cornices,  architraves,  or  cabinet 
mouldings.  The  method  of  working  it  out  is  so  obvious  as  to  re- 
quire no  further  explanations. 


PRACTICAL  CARPENTRY. 


83 

A method  is  here  shown  of  mitering  a gothic  capital  or  base 
round  a square  standard,  a,  Fig.  129,  shows  the  standard,  s s s s 


show  the  blocks  that  form  the  capital  or  base.  The  two  other  fig- 
ures,  130  and  131,  show  the  base  and  capital  completed. 

Sometimes  there  are  more  than  four  pieces  in  the  cluster,  but  the 
same  principles  rule  when  such  is  the  case;  the  central  standard, 
however,  should  have  as  many  sides  on  it  as  there  are  pieces  in  the 
cluster. 

Sometimes  the  workman  will  find  it  necessary  to  mitre  in  circles 
between  two  lines  of  mouldings,  and  to  do  so  the  circular  mould- 
ings must  be  made  with  a diameter  large  enough  to  have  the  solid 


*4 


PRACTICAL  CARPENTRY. 


Fig.  130. 


Fig.  131. 


adjoin  the  solid  wood  of  the  running  mouldings,  as  shown  at 


Fig.  132.  It  will  be  seen 
that  the  points  of  juncture 
of  the  various  members  of 
the  mouldings  do  not  run  in 
a straight  line,  thus  making 
the  mitre-joint  a little  curved, 
which  admits  of  the  mould- 
ings working  together  accu- 
rately without  requiring  to 
be  pared. 

Figs.  133,  S34,  135  and 
136  show  how  various  joints 
are  made  by  the  junction  of 
circular  mouldings  with 
straight  mouldings,  and 
mouldings  'vdth  more  or  less 
curvature. 

A “spring”  moulding  is 
one  that  is  made  of  thin 
stuff,  and  is  leaned  over  to 
make  the  proper  projection, 
as  shown  at  Fig.  138.  a 
shows  the  spring  moulding; 
b the  space  left  vacant  by 


Fig.  m 


rei  KI  -3m  -Stj 


PRACTICAL  CARPENTRY. 


PRACTICAL  CARPENTRY. 


PRACTICAL  CARPENTRY. 


8? 


the  leaning  of  the  moulding.  These  mouldings  are  difficult 
to  mitre,  more  particularly  so  when  the  joint  is  made  with 
a raking  moulding  that  “ springs  ” also.  Some  of  the  methods 
given  for  obtaining  the  cuts  for  raking  mouldings  may  be  used 
for  cutting  these  mouldings  when  the  work  is  straight;  bu 
when  circular,  the  application  of  other  methods  is  sometimes 
necessary.  Many  times  the  workman  will  come  across  very 
knotty  operations  of  this  kind  to  work  out,  and  the  following 
diagrams  will  then  prove  exceedingly  useful:  Fig.  137  exhibits  an 

elevation  of  a circular  moulding  mitered  into  a horizontal  mould- 
ing. The  shape  and  plane  of  the  moulding  is  shown  at  b.  It  is 
evident  that  by  producing  the  line  ad  to  intersect  the  centre  line 
of  the  arc  at  c,  the  central  point  will  be  obtained,  from  which  the 
circular  piece  required  for  the  moulding  may  be  described,  e a 
and  f d give  the  radius  for  the  curves  of  both  edges  when  the  stuff 
is  in  position,  as 
shown  in  the  ele- 
vation. 

Fig.  139  shows 
the  application  of 
the  same  rules  to  a 
circular  elevation 
of  a different  form 
standing  over  a 
straight  plan.  The 
back  lines  of  the 
moulding  are  produced  until  th^y  bisect  a horizontal  line  drawn 
through  the  centre,  from  which  the  circular  cornice  was  struck,  as 
shown  by  the  lines  a b and  c d.  In  other  respects  the  operation  is 
precisely  the  same  as  at  Fig.  137. 


PART  Yltt— JOIEERY. 


|raP^O  vnake  a good  dovetail  joint  properly  requires  considerable 
sHll  and  care  on  the  part  of  the  operator;  but  when 
completed,  no  other  system  of  joining  boards  at  right 
angles  proves  so  satisfactory.  This  joint  has  three  varieties:  is\ 

the  common  dovetail,  where  the  dovetails  are  seen  on  each  side  of 
the  angle  alternately;  2d,  the  lapped  dovetail,  in  which  the  dove- 
tails are  seen  only  on  one  side  of  the  angle ; and,  3d,  the  lapped  and 
mitred  dovetail,  in  which  the  joint  appears  externally  as  a common 
mitre-joint.  The  lapped  and  mitred  joint  i?  useful  in  salient  angles. 


No.  1.  No.  % No.  S. 


Fig.  140. 


m finished  work,  but  it  is  not  so  strong  as  the  common  dovetail, 
and  therefore,  in  all  re-entrant  angles,  th  e latter  should  be  used. 
The  three  varieties  are  shown  in  the  annexed  engravings. 

Fig.  140,  No.  1,  shows  the  open  dovetail,  and  No.  2 is  a 
perspective  view  of  the  same  showing  how  the  work  appears 
whe&  ready  to  put  together.  No.  3 shows  the  work  com- 


PRACTICAL  CARPENTRY. 


89 


plete.  Fig.  141,  Nos.  1 and  2,  show  the  lapped  dovetail.  This 
style  is  generally  used  for  drawers  and  work  where  it  is  de- 
sirable that  end  wood  should  not  be  seen  in  the  front  side. 
No.  3 shows  the  angle  and  position  of  lines  of  juncture  when  put 
together.  Fig.  142,  No.  1,  2,  3 and  4,  show  the  method  of  making 
a combined  dovetail  and  mitre  joint.  This  style  of  joint  is  some- 
times called  a “ blind”  dovetail-joint;  it  is  chiefly  used  in  nvT.ng 


fancy  boxes  where  all  the  corners  are  exposed.  The  joint  shotfs  a 
mitre  on  the  outside,  the  pins  and  mortises  of  the  dovetails  being 
hidden  from  view. 

Figs.  143  and  144,  show  two  methods  of  dovetailing  hoppers, 
trays  and  other  splayed  work.  The  reference  letters  a and  b show 
that  when  the  work  is  together  a will  stand  directly  over  b.  Care 
must  be  taken  when  preparing  the  ends  of  stuff  for  dovetailing,  for 
hoppers,  trays,  etc.,  that  the  right  bevels  and  angles  are  obtained^ 
according  to  the  rules  explained  for  finding  the  cuts  and  bevels  for 
hoppers  and  works  of  a similar  kind,  in  the  examples  given  further 
<M. 

Figs.  145,  146 and  147,  show  how  to  get  the  “cuts”  or  bevels  for 
hoppers.  It  is  old,  but  correct  and  simple.  Let  Fig.  143  represent 


PRACTICAL  CARPENTRY. 


ifO 


lie.  I Wo.  2, 


Fi$  142: 


PRACTICAL  CARPENTRY. 


91 


No.  L No.  2. 


No.  1.  No.  2. 


a box  whose  sides  splay,  or  we  may  call  it  a hopper  or  breud-tmy ; 
or  it  may  form  the  back  and  two  ends  of  a carnage  se&t,  if  we  take 
off  one  side  and  cut  it  off  at  the  dotted  lines,  where  a seat  would 
be  required.  The  lines  n r show  the  bevel  or  splay  of  the  box. 
Take  this  bevel  and  transfer  it  to  the  end  of  one  piece  of  the  stuff 
you  intend  using,  as  shown  at  p,  Fig.  146.  Then  take  the  distance 
s o,  on  the  marking  gauge,  and  run  a line  on  the  board  from  o to 
k,  Fig.  147  Now  make  the  line  a b the  same  bevel  as  the  hop- 
per, or,  the  bevel  as  now  set,  may  be  applied  with  the  stock  resting 


PRACTICAL  CARFS^riY. 


m ihe  edge  r,  then  the  blade  will  represent  the  line  a b.  Square 

up  from  m,  cut- 
ting a r at  p, 
then  square  over 
on  the  edge  the 
line  pe.  Then 
connect  a e, 
which  will  be  the 
angle  required 
for  a butt  joint, 
the  inside  comer 
being  the  long- 
est. If  it  is  re- 
quired to  mitre 
the  joint,  set  off 
the  thickness  of 

the  stuff  as  measured  at  z o,  from  a to  x,  then  the  line  x e will  be 
the  bevel  sought. 

Another  method  for  accomplishing  the  same  result  is  given 
below.  This  method  in  many  shapes  and  forms,  has  been  used 
from  time  immemorial  by  workmen,  more  particularly  by  car- 
riage makers  to  obtain  the  bevels  of  splayed  seats;  the  present 
way  of  expressing  it,  however,  is  comparatively  recent. 

* If  we  make  a i,  Fig.  148,  represent  the  elevation  of  our  hop- 

* Fro®  “The  Steel  8quare  and  Its  Uses.” 


E 


Fig.  147. 


PRACTICAL  CARPENTRY, 


95 


f^r,  a.id  3 i a portion  of  the  plan,  we  proceed  as  follows*  Lay  off 
S,  whicli  is  the  bevel  of  one  side,  and  spo  the  section  of  o ne  end> 
Place  one  foot  of  the  dividers  where  the  bevel  is  shown,  and  with 
s as  radius  describe  the  arc  s u,  intersecting  the  right  line  u in  the 
point  u.  At  s erect  the  perpendicular  s t,  and  draw  the  line  u t at 
right  angles  to  cu.  Connect  t with  centre ; then  the  triangle  st  is 
the  end  bevel  required.  The  line  x is  the  hypothenuse  of  a right 
angled  triangle,  of  which  u may  be  taken  for  the  perpendicular. 


and  u t for  the  base.  To  find  the  mitre  of  which  d e is  the  plan, 
project  s and  p,  as  indicated  in  the  plan  by  the  full  lines.  With  s 
p as  radius  and  s as  centre,  describe  the  arc  p r.  In  the  plan  draw 
d g,  on  which  lay  off  the  distance  s r,  measuring  from  f,  as  shown 
by  ?c.  Then  o k f is  the  mitre  sought. 

Fig.  149  shows  the  rule  for  finding  the  bevels  for  the  sides  of  the 
hopper.  From  M,  the  point  at  which  em  intersects  u c.  or  the 
inner  face  of  the  hopper,  erect  the  perpendicular  m l.  intersecting 


94 


i KACTro&c. 


r r,  or  the  upper  edge  of  the  hopper,  in  the  point  L.  Then  L c 
shows  how  much  longer  the  inside  edge  is  required  to  be  than  the 
outside.  In  the  plan  draw  tv  parallel  to  sux,  making  the  distance 
between  the  two  lines  equal  to  c f of  the  elevation,  or,  equal  to  the 
thickness  of  one  side.  From  the  point  l in  the  elevation  drop  the 
line  l w,  producing  it  until  it  cuts  the  mitre  line  n o,  as  shown  at 


v.  From  w.  at  right  angles  to  l w,  erect  the  perpendicular  w v, 
meeting  the  line  x v in  the  point  v.  Connect  v and  u ; then  tvu 
will  be  the  angle  sought.  This  bevel  may  be  found  at  once  by 
laying  off  the  thickness  of  the  side  from  the  line  e m,  as  shown  by 
n p in  the  elevation,  and  applying  the  bevel  as  shown.  This  course 
does  away  with  the  plan  entirely,  provided  both  sides  have  the  same 
inclination. 

There  are  several  other  ways  by  which  the  same  results  may  be 
obtained  ; some  of  these  will  no  doubt  occur  to  the  reader  when 
laying  out:  the  lines  ns  shown  ere. 


PRACTICAL  CARPENTRY. 


PAPiT  IX.  ’MISCELLANEOUS  PROBLEMS. 

ENT  WORK:  Fig.  150  exhibits  a method  of  obtaining  the 
correct  shape  of  a veneer  for  covering  the  splayed  head  of 
a gothic  jamb,  e shows  the  horizontal  sill,  e f the  splay, 
/ a the  line  of  the  inside  of  jamb,  0 the  difference  between  front  and 
back  edges  of  jamb,  b a the  line  of  splay.  At  the  point  of  junction  of 
the  lines  b a, /a,  set 
one  point  of  the 
compasses,  and  with 
the  radius  a b draw 
the  outside  curve  of 
n ; then  with  the  ra- 
dius a s draw  the 
inside  curve,  and  n 
will  be  the  veneer 
required.  This  'will 
give  the  required 
shape  for  either  side 
of  the  head. 

Developing  Cylin- 
ders —Cylinders  may 
be  considered  as 
prisms,  of  which  the 
base  is  composed  of  an  infinite  number  of  sides.  Thus  we  shall 
obtain  graphically  the  development  of  a right  cylinder  by  a rect- 
angle of  the  same  height,  and  of  a length  equal  to  the  circumfer- 
ence of  the  circle,  which  serves  as  its  base,  measured  by  a greater 
or  lesser  number  of  equal  parts. 

But  if  the  cylinder  (Fig.  151)  be  oblique,  and  it  is  required  to 
diaw  its  profile  as  inclined,  describe  on  the  centre,  of  the  axis  of  tne 
inclined  profile,  and  perpendicular  to  it,  the  circle  or  ellipse  which 


96 


PRACTICAL  CARPENTRY. 


forms  the  base ; and  divide  its  circumference  into  a number,  of  equal 
parts,  and  through  these  divisions  draw  lines  parallel  to  the  axis  a 
b,  c d,  e f g h,  etc. 

Then  to  find  the  projection  of  the  base  on  a horizontal  plane, 

from  the  points  ace 
g,  where  the  lines 
from  the  divisions  of 
the  circumference 
meet  the  line  of  the 
base  <2  b,  let  fall  per- 
pendiculars on  a Sine 
a!  k\  parallel  to  the 
base,  and  produce 
them  indefinitely  be- 
yond it.  From  the 
points  m'  ri  o'  p\ 
where  these  perpen- 
diculars intersect  the 
line  a!  k\  set  off  on 
each  side  m'  x,  mf  15, 
and  n ' 2,  nr  14,  equal  to  the  ordinates  of  the  circle  distinguished  by 
the  same  letters  and  figures,  and  so  on  with  the  other  divisions ; and 
through  the  points  thus  obtained,  draw  the  ellipse  a , 4,^,  1 2,  which 
is  the  projection  of  the  base  of  the  cylinder  on  a horizontal  plane. 

To  obtain  the  development  of  the  cylindrical  surface,  produce  e 
r indefinitely  to  g,  and  set  out  on  it  from  e'  the  divisions  of  the  cir- 
cumference of  the  circle  1234,  etc.,  in  the  points  in  n o /,  etc.; 
through  these,  draw  lines  parallel  to  the  axis,  and  transfer  to 
them  the  lengths  of  the  corresponding  divisions  of  the  profile, 
as  e a,  e b , m c , m d,  n <?,  11  f etc.;  then  draw  the  curves  a c eg  H, 
bdfh  a,  through  the  points  thus  obtained.  The  addition  of  the 
elliptic  surfaces,  which  form  the  base  and  head  of  the  solid,  and 
which  are  similar  and  equal  to  a\  4,  k ' 12,  completes  the  develop 
ment. 

The  extent  e'  g will  not  be  truly  the  same  as  that  of  the  periph- 
ery <if  the  circle  ft  f,  inasmuch  as  the  distances  in  the  !at*ei  are  but 


PRACTICAL  CARPENTRY. 


97 


the  cords  of  segments;  if,  however,  the  number  of  divisions  em- 
ployed be  ample,  the  amount  of  the  error  will,  for  practical  pur- 
poses, be  inappreciable. 

Taking  .Dimensions. — In  taking  the  dimensions  of  any  triangular 
figure,  make  a sketch  of  it  as  in  Fig.  152,  No.  1,  and  on  each  line 


of  the  sketch  mark  the  dimensions  of  the  side  of  the  figure  it  repre- 
sents. Then,  in  describing  the  figure,  either  to  its  full  dimensions, 
or  to  any  proportionate  scale,  draw  any  straight  line  as  ab,  No.  2 
and  make  it  equal  to  the  dimension  marked  on  the  corresponding 
line  a b of  the  sketch  No.i.  From  the  centre  a and  With  the  radius 
ac,  describe  an  arc  at  c;  then  from  the  centre  b,  with  the  radius  B 
c,  describe  an  arc  intersecting  the  former : join  a c,  b c,  and  the 
triangle  a c b is  the  figure  required. 

The  dimensions  of  any  figure  are  taken  on  the  principle  above 


illustrated.  If  the  figure  is  not  triangular,  it  is  divided  into  tii 
rmgles,  in  the  manner  shown  by  Fig.  153,  Nos.  1 and  2. 

In  Fig.  154,  Nos.  1 and  2,  the  manner  of  taking  dimensions, 
when  one  or  more  sides  of  the  figure  are  bounded  by  curved  lines, 


93 


PRACTICAL  CARPENTRY. 


is  illustrated.  When,  as  at  a b (No.  i),  the  side  is  a circular  arc, 
its  centre  is  obtained  as  follows:  The  extreme  points  a b,  and  the 

point  of  junction  c of  the  intermediate  line  e c with  a c and  b c, 
give  three  points  in  the  curve.  From  a and  b,  therefore,  with  any 


radius,  describe  arcs  above  and  below  the  curve ; from  c,  with  the 
same  radius,  intersect  these  arcs;  through  the  intersections  draw 
straight  lines  meeting  in  d ; and  d is  the  centre  of  th^  curve,  and 
E A*  D b,  or  d c its  radius. 


PRACTICAL  CARPENTRY* 


99 


PART  X.— TIMBER  WORK 


01  NTS  AND  STRAPS. — Plate  I.  shows  a number  of 
joints  in  framing  that  will  frequently  be  found  useful  by  the 
workman.  Fig.  i,  No.  i,  shows  the  joint  formed  by  the 
meeting  of  a principal  rafter  and  tie-beam,  c being  the  tenon.  The 
cheeks  of  the  mortise  are  cut  down  to  the  line  d f so  that  an  abut- 
ment, e d9  is  formed  of  the  whole  width  of  the  cheeks,  in  addition 
to  that  of/the  tenon;  and  the  notch  so  formed  is  called  a joggle. 
No.  2 shows  the  parts  detached  and  in  perspective,  and  it  will  be 
seen  that  a much  larger  bearing  surface  is  thus  obtained. 

Fig.  2. — No.  i is  a geometrical  elevation  of  a joint,  differing  from, 
the  last  by  having  the  anterior  part  of  the  rafter  truncated,  and  the 
shoulder  of  the  tenon  returned  in  front.  It  is  represented  in  per- 
spective in  No.  2. 

Fig.  3. — Nos.  1 and  2 show  the  geometrical  elevation  and  pei- 
spective  representation  of  an  oblique  joint,  in  which  a double  abut- 
ment or  joggle  is  obtained.  In  all  these  joints,  the  abutment,  as  d 
e9  Fig.  1,  should  be  perpendicular  to  the  line  d /;  and  in  execution, 
the  joint  should  be  a little  free  at  /,  in  order  that  it  may  not  be 
thrown  out  at  d by  the  settling  of  the  framing.  The  double  abut- 
ment is  a questionable  advantage;  it  increases  the  difficulty  of  exe- 
cution, and,  of  course,  the  evils  resulting  from  bad  fitting.  It  is 
properly  allowable  only  where  the  angle  of  meeting  of  the  timbers 
is  very  acute,  and  the  bearing  surfaces  are  consequently  very  long. 

Fig.  4.— Nos.  1 and  2 show  a means  of  obtaining  resistance  to 
sliding  by  inserting  the  piece  c in  notches  formed  in  the  rafter  and 
the  tie-beam  : d e shows  the  mode  of  securing  the  joint  by  a bolt. 

Fig.  5. — Nos.  1 and  2 show  a very  good  form  of  joint,  in  which 
the  place  of  the  mortise  is  supplied  by  a groove  c in  the  rafter,  and 
the  place  of  the  tenon  by  a tongue  d in  the  tie-beam.  As  the  part? 
can  be  all  seen,  they  can  be  more  accurately  fitted,  which  is  an  ad 


too 


PRACTICAL  CARPENTRY. 


vantage  in  heavy  work.  In  No.  i the  mode  of  securing  the  joint 
by  a strap  a b and  bolts  is  shown. 

Fig.  6. — Nos.  i and  2 is  another  mortise-joint,  secured  by  a 
strap  a b and  cotter  or  wedge  a. 

Fig.  7 shows  the  several  joints  which  occur  in  framing  the  king- 
post into  the  tie-beam,  and  the  struts  into  the  king-post,  a is  the 
tie-beam  ; b,  the  king-post;  and  c and  d,  struts.  The  joint  at  the 
bottom  of  the  king-post  has  merely  a short  tenon  e let  into  a mor- 
tise'in  the  tie-beam.  The  abutment  of  the  strut  d is  made  square 
to  the  back  of  the  strut,  as  far  as  the  width  of  the  king-post  admits, 
and  a short  tenon  /is  inserted  into  a mortise  in  the  king-post.  The 
abutment  of  the  joint  of  c is  formed  as  nearly  square  to  the  strut  as 
possible. 

The  term  king-post,  as  has  been  already  stated,  gives  quite  an 
erroneous  notion  of  its  functions,  which  are  those  of  a suspension 
tie.  Hence  the  necessity  for  the  long  strap  b a bolted  at  d dt  and 
secured  by  wedges  at  c,  in  the  manner  more  distinctly  shown  by  the 
section,  Fig.  8,  No.  2.  The  old  name  king-piece  is  better  than 
king-post. 

Fig.  8. — No.  1 shows  the  equally  inappropriately  named  queen- 
post . a is  the  tie-beam;  b,  the  post  tenoned  at  e;  c,  the  strut; 
and  d,  the  straining-piece.  The  strap  ba,  and  bolts  d d. 

Fig.  9. — In  this  figure,  the  superior  construction  is  shown,  in 
which  a king-bok  of  iron  c d is  substituted  for  the  king-post.  On 
the  tie-beam  a,  is  bolted  by  the  bolts  a e,  dp  the  cast-iron  plate 
and  sockets  abed , the  inner  parts  of  which,  Ji  g , h g,  form  solid 
abutments  to  the  ends  of  the  struts  b b.  The  king-bolt  passes 
through  a hole  in  the  middle  of  the  cast-iron  socket-plate,  and  is 
seared  below  by  the  nut  d.  A bottom-plate  e f prevents  the 
crushing  of  the  fibres  by  the.  bolts. 

Plate  II. — Figs.  1 to  5 show  various  methods  of  framing  the 
head  of  the  rafters  and  king-posts  by  the  aid  of  straps  and  bolts. 
Fig.  6 shows  the  heads  of  the  rafters  halved  and  bolted  at  their 
junction,  and  a plate  laid  over  the  apex  to  sustain  the  .bolts  which 
are  substituted  for  the  king-post.  One  bolt  necessarily  has  a link 
formed  in  it  for  the  other  to  pass  through. 


PRACTICAL  CARPENTRY. 


lOI 


Fig,  7 shows  at  d what  may  be  considered  the  upper  part  A the 
same  king  bolt  as  is  shown  in  Plate  I.,  Fig.  9,  with  the  mode  of 
connecting  the  rafters.  A cast-iron  socket-piece  c receives  the 
tenons  a a of  the  rafters  a a,  and  has  a hold  through  it  for  the  bolt, 
the  head  of  which,  b , is  countersunk,  b is  the  ridge-piece  set  in  a 
shallow  groove  in  the  iron  socket-piece.  An  elevation  of  the  side 
is  given,  in  which  g is  the  bolt,  f the  socket-piece,  and  e the  ridge- 
piece. 

Figs.  8,  9,  10  and  11  illustrate  the  mode  of  framing  together  the 
principal  rafter,  queen-post,  and  straining-piece.  In  the  first  three 
examples  the  joints  are  secured  by  straps  and  bolts;  and  in  the 
last  example  the  queen-bolt  d passes  through  a cast-iron  socket- 
piece  c,  which  receives  the  ends  of  the  straining  piece  and  rafter,  as 
those  of  the  two  rafters  are  received  in  Fig.  7. 

Figs.  12  and  13  show  modes  of  securing  the  junction  of  the 
collar-beam  and  rafter  by  straps;  and  Figs.  14  and  15,  modes  of 
securing  the  junction  of  the  strut  and  the  rafter  by  straps. 

Lengthening  Beams,  etc. — In  large  works  in  carpentry  it  is  often 
necessary  to  join  timbers  in  the  direction  of  their  length,  in  order 
to  get  them  long  enough  to  answer  the  purpose.  When  it  is  neces- 
sary to  maintain  the  same  depth  and  width  in  the  lengthened 
beam,  the  mode  of  joining  called  scarfing  is  employed.  Scarfing  is 
performed  in  a variety  of  ways,  dependent  upon  whether  the  length- 
ened beam  is  to  be  subjected  to  a longitudinal  or  transverse  strain. 
This  method  of  joining  is  illustrated  in  Plate  III.,  Figs.  1 to  13. — 
The  methods  are  self  evident. 


109 


PRACTICAL  CARPENTRY. 


PART  XL— SWING  JOINTS. 


RINGING. — Plate  IV.  shows  a number  of  methods  of 
hinging.  Fig.  i,  No.  i,  shows  the  hinging  of  a door  to 
open  to  a right  angle,  as  in  No.  2. 

Fig.  2,  Nos.  1 and  2,  and  Fig.  3,  Nos.  1 and  2.  These  figures 
show  other  modes  of  hinging  doors  to  open  to  90°. 

Fig.  4,  Nos.  1 and  2.  These  figures  show  a manner  of  hinging 
a door  to  open  to  90°,  and  in  which  the  hinge  is  concealed.  The 
segments  are  described  from  the  centre  of  the  hinge  gy  and  the 
dark  shaded  portion  requires  to  be  cut  out  to  permit  it  to  pass  the 
leaf  of  the  hinge  g f 

Fig.  5,  Nos.  1 and  2,  show  an  example  of  centre-pin  hinge  per- 
mitting the  door  to  open  either  way,  and  to  fold  back  against  the 
wall  in  either  direction.  Draw  a b at  right  angles  to  the  doof,  and 
just  clearing  the  line  of  the  wall,  or  rather  representing  the  plane  in 
which  the  inner  face  of  the  door  will  lie  when  folded  back  agains; 
the  wall;  bisect  it  inland  draw  f d the  perpendicular  to  aby  which 
make  equal  to  af  or f by  and  d is  the  place  of  the  centre  of  the  hinge. 

Fig.  6,  Nos.  1 and  2,  another  variety  of  centre-pin  hinging  open- 
ing to  90°.  The  distance  of  b from  a c is  equal  to  half  of  a c . In 
this,  as  in  the  former  case,  there  is  a space  between  the  door  and 
the  wall  when  the  former  is  folded  back.  In  the  succeeding  figures 
this  is  obviated. 

Fig.  7,  No.  1.  Bisect  the  angle  at  a by  the  line  a b , draw  d e 
and  make  eg  equal  to  once  and  a half  times  ad;  draw  fg  at  right 
angle  to  cd , and  bisect  the  angle  fge  by  the  line  cgy  meeting  a b 
in  b , wrhich  is  the  centre  of  the  hinge. 

No.  2 shows  the  door  folded  back  when  the  point  e falls  on  the 
continuation  .of  the  line  f g. 

Fig.  8,  Nos.  1 and  2 To  find  the  centre  draw  aby  making  an 
angle  of  450  with  the  inner  edge  of  the  door,  and  draw  c b parallel 


PRACTICAL  CARPENTRY. 


*°3 

to  the  jamb,  meeting  it  in  b , which  is  the  centre  of  the  hinge.  The 
door  revolves  to  the  extent  of  the  quadrant  d c . 

Pl.ate  V. — Fig.  i,  Nos.  i and  2;  Fig.  2,  Nos.  1 and  2;  and 
Fig.  3,  Nos.  1 and  2,  examples  of  centre-pin  joints,  and  Fig.  4,  Nos. 
1 and  2,  do  not  require  detailed  description. 

Fig.  5,  Nos.  1,  2,  and  3,  show  the  flap  with  a bead  <2  closing  into 
a corresponding  hollow,  so  that  the  joint  cannot  be  seen  through. 

Fig.  6,  Nos.  1,  2,  and  3,  show  the  hinge  a b equally  let  into  the 
styles,  and  its  knuckle  forming  a part  of  the  bead  on  the  edge  of 
the  style  b.  The  beads  on  each  side  are  equal  and  opposite  to 
each  other,  and  the  joint  pin  is  in  the  centre. 

Fig.  7,  Nos.  1,  2,  and  3.  In  this  example,  the  knuckle  of  the 
hinge  forms  portion  of  the  bead  on  the  style  b,  which  is  equal  and 
opposite  to  the  bead  on  the  style  a. 

In  Fig.  8,  Nos.  1,  2,  and  3,  the  beads  are  not  opposite. 

Plate  VI. — Fig.  1,  shows  the  hinging  of  a back  flap  when  the 
centre  of  the  hinge  is  in  the  middle  of  the  joint. 

Fig.  2,  Nos.  1 and  2,  shows  the  manner  of  hinging  a back  flap 
when  it  is  necessary  to  throw  the  flap  back  from  the  joint. 

Fig.  3,  Nos.  1 and  2,  is  an  example  of  a rule-joint,  such  as  is  re- 
quired for  the  shutter  ,s.  The  further  the  hinge  is  imbedded  in  the 
wood,  the  greater  will  be  the  cover  of  the  joint  when  opened  to  a 
right  angle. 

Fig.  4,  Nos.  1 and  2,  shows  the  manner  of  finding  the  rebate 
when  the  hinge  is  placed  on  the  contrary  side. 

Let  /be  the  centre  of  the  hinge,  a b the  line  of  joint  on  the  same 
side,  h c the  line  of  joint  on  the  opposite  side,  and  b c the  total 
depth  of  the  rebate.  Bisect  be  in  e and  join  e f;  on  <f/ describe  a 
semicircle  cutting  a b in  g , and  through  g and  e draw  g h cutting  d 
c in  //,  and  join  d h , hg , and  ga  to  form  the  joint. 

Fig.  5,  Nos.  1 and  2,  is  a method  of  hinging  employed  when  the 
flap  on  being  opened  has  to  be  at  a distance  from  the  style.  It 
is  used  in  doors  of  pews  to  throw  the  opened  flap  or  door  clear 
of  the  mouldings  of  the  coping. 

Fig.  6,  Nos.  1 and  2,  is  the  ordinary  mode  of  hinging  the 
shutter  to  the  sash  frame. 


104 


PRACTICAL  CARPENTRY. 


PART  X1T.— USEFUL  RULES,  TABLES, 
DATA  AND  MEMORANDA 

ENTRES. — A few  remarks  on  the  subject  of  centres  may 
prove  of  use  to  the  carpenter  and  joiner,  as  he  frequently 
has  to  prepare  and  fix  them  in  place.  I have  given  a 
few  general  - directions  in  preceding  pages,  but  not  sufficient  to 
satisfy  every  requirement ; therefore,  a few  observations  here  will 
not  be  untimely. 

Centres  are  temporary  structures  of  wood,  with  curved  upper 
surfaces  upon  which  arches  are  built,  and  left  until  they  are  con- 
solidated and  have  taken  their  bearing,  after  which  the  centres  are 
removed. 

For  large  arches,  such  as  those  of  bridges,  very  elaborate  centres 
are  required,  with  special  arrangements  for  easing  and  striking 
them  gradually;  but  in  ordinary  buildings  the  centres  are  very 
simple;  the  arches  for  which  they  are  required  being  generally  of 
small  span  and  common  construction. 

Centres  for  very  small  and  narrow  arches  may  consist  simply  of 
a piece  of  wood  cut  to  the  curve  of  the  soffit  of  the  arch,  and  sup- 
ported under  it  by  uprights. 

For  longer  arches, *such  as  those  of  tunnels,  sewers,  etc.,  the 
centre  is  composed  of  a number  of  curved  pieces  or  ribs,  and  have 
narrow  battens  or  lagging  nailed  across  them. 

For  stone  arches  or  very  rough  brick  arches,  the  battens  may  be 
placed  from  one  to  two  inches  apart,  but  for  superior  brick  arches 
they  should  be  placed  close  together,  and  the  arrises  or  comers 
taken  off  with  a rough  jack  plane.  As  few  nails  as  possible  should 
be  used  in  putting  on  the  lagging,  in  order  to  save  trouble  when 
the  centering  is  done  with  and  being  removed. 

When  the  arches  are  small,  single  pieces  may  form  the  centre ; 


PRACTICAL  CARPENTRY. 


I05 

and  when  the  opening  does  not  exceed  six  or  eight  feet,  the  centre 
may  be  built  up  of  several  layers  of  thin  boards,  with  the  joints 
broken,  and  the  whole  well  nailed  together.  More  elaborate 
centres,  for  bridges,  large  openings  in  warehouses,  factories  or  tun- 
nels, should  be  built  of  timber,  properly  framed  together  in  such  a 
manner  that  they  may  be  taken  down  and  apart  without  cutting 
the  timber  of  which  they  are  composed. 

In  constructing  all  but  the  very  smallest  arches  arrangements 
should  be  made  for  easing  the  centres,  so  as  to  gradually  deprive  the 
arches  of  their  supports.  This  is  done  by  means  of  pairs  of  greased 
wedges  introduced  between  the  heads  of  the  supports  and  the  rest- 
ing points  of  the  centres. 

After  the  arch  has  been  turned,  and  the  haunches  filled  in,  the 
points  of  the  wedges  may  be  lightly  struck  with  a hammer  so  as  to 
drive  them  outwards  from  the  rib  under  which  they  are  placed, 
thus  lowering  the  centre  a very  little ; which  causes  the  whole  of  the 
arch  to  settle  slightly  and  uniformly,  and  to  take  its  bearing,  the 
mortar  being  compressed  in  the  joints.  The  arch  is  then  left  until 
the  mortar  has  set,  after  which  the  centres  are  removed  altogether. 
Some  builders  defer  the  easing  of  the  centres  for  a day  or  two  after 
the  arch  is  finished.  When  an  arch  is  built  of  soft  stone  or  bricks, 
and  the  super-incumbent  weight  is  great,  it  is  better  not  to  ease  the 
centering  for  some  time,  as  the  pressure  on  the  edges  of  the  vous- 
soirs  is  apt  to  crack  and  chip  them.  In  centres  for  very  important 
stone  arches,  wedges  or  screws  are  frequently  placed  under  each 
piece  of  lagging,  so  that  the  work  may  be'  eased  “ course  by 
course,”  and  the  supports  wedged  up  again  if  the  settlement  is  too 
rapid.  Arrangements  are  sometimes  adopted  for  easing  all  the 
wedges  at  the  same  time,  so  that  the  whole  arch  may  settle  uni- 
formly, but  this  is  complicated  and  expensive,  and  on  these  ac- 
counts will  never  become  popular. 

Seasoned  materials  should  always  be  employed  for  centres  in- 
tended for  the  better  sort  of  arches. 

Estimates.— It  frequently  happens  that  the  carpenter  is  called 
upon  to  make  estimates  of  buildings,  more  particularly  of  buildings 
of  the  cheaper  sort,  and  in  order  to  do  this  intelligently  and  with 


io6 


PRACTICAL  CARPENTRY. 


any  degree  of  accuracy,  he  should  be  able  to  figure  up  the  dimen- 
sions of  all  the  materials  used  in  and  about  the  building,  and  to, 
also,  know  the  cost  of  the  various  materials  delivered  on  the  site, 
the  exact  price  of  the  various  kinds  of  labor,  and  have  a thorough 
knowledge  of  the  details  of  his  own  trade,  and  a fair  knowledge  of 
the  practical  operations  of  the  other  trades  that  will  be  employed 
on  the  building  estimated  for.  This  knowledge  is  necessary  in 
order  to  provide  for  unseen  contingencies  which  frequently  arise 
in  building,  and  which  are  often  sources  of  annoyances  to  both 
builders  and  owners.  A few  hints  are  here  given  which  may  aid 
the  estimator,  and  for  further  information  on  this  subject  I refer 
him  to  the  Builders  Guide  and  Estimator's  Price  Book , as  it  con- 
tains rules,  prices  of  labor  and  material,  costs  of  manufactured 
stuff,  such  as  sashes,  doors,  stairs,  hardware,  glass,  paints,  plumber’s 
goods,  etc. 

The  following  is  a form  for  an  estimate;  it  may  be  ruled  on  fools- 
cap paper,  and  when  the  blank  columns  are  properly  filled  out,  and 
the  estimate  completed,  it  should  be  filed  away  for  future  reference, 
and  if  any  errors  or  mistakes  have  been  made,  corrections  should  be 
made  on  the  margin  with  memoranda  attached,  noting  how  the 
mistakes  occurred.  The  form  is  only  intended  as  a guide,  and  there- 
fore only  includes  a few  of  the  items  required  about  a building. 
It  must,  of  course,  be  changed  to  suit  the  requirements  of  the 
building  under  consideration : 


FORM  FOR  AN  ESTIMATE. 


Quantities.  Description  of  Work. 

Price. 

$ 

250  yards  cube.  Excavating  for  foundation  walls,  drains, 

o a!  a nf  a o n rl  VPlTinVlTUy  fit  HIT  . , , 

POSIS,  ClC.y  ClbM  CUJII  lCUlUVlU^  

36  Cords  Rubble  masonry  in  foundation  walls,  including 
all  material  set  in  mortar,  pointed  and  complete,  in- 

/'»']  n r"l  i rr  romnvinfT  rilhhifih  

80  Thousand  brick  laid  in  mortar,  in  walls  and  parti- 
tions, joints  struck  including  setting  of  all  walls, 

o t ae  V»rkQT*.Ic  n rt rl  ntbpr  timber  

TJlrilCo,  UUdl  Uo  HI  Hi  uiuci  iiui  i/ci  . •.••••• 

2500 Feet,  lineal,  of  flooring  joists,  11  x 3,  fixed  complete 

oil  f nviYi i n or  niPFPR  etc  ..  

WltLl  Ull  lllliniinig  fjicocn,  ciu 

15  Squares  of  1£  inch  matched  flooring  laid  complete, 

PRACTICAL  CARPENTRY. 


107 


form  for  an  estimate  (Continued). 


Quantities.  Description  of  Work. 

Price. 

$ 

c. 

16  Doors,  7 x 2 TO,  2 inches  thick,  in  four  panels,  moulded 
on  both  sides,  with  7 inch  moulded  architraves, 
easing  and  saddle,  including  a lock  worth  75  cents, 
and  3 inch  butt  hinges;  hung  and  fixed  complete. . . 
10  Sashes  and  frames,  sashes  6 ft.  by  3.6  in  six  lights, 
glazed  (state  kind  of  glass),  best  delf  knobs,  wroaght 

iron  bolts,  hinges,  and  fixed  complete 

10  Sets  of  inside  shutters  panelled,  to  open  in  two 
pieces  with  flaps,  panelled  and  moulded  soffit,  in- 
cluding sills,  etc.,  all  fixed  complete 

10  Double  windows,  describe 

10  sets  of  summer  blinds  and  frames,  each  side  to  open 

in  two,  with  moveable  slate,  fixed  all  complete 

950  Yards  superficial  of  plastering  in  two  coats  and  set 
with  fine  stuff,  including  lathing  and  all  materials. . 
Cornices 

Centre-pieces 

The  foregoing  is  lengthy  enough  to  give  the  intelligent  work- 
man a clear  idea  of  the  proper  method  of  making  an  estimate. 

To  make  the  matter  clearer,  however,  the  following  general  mem- 
oranda of  items  are  given,  so  as  to  show  how  the  work  should  be 
described  or  itemized  in  order  to  avoid  omissions  and  mistakes: 

GENERAL  MEMORANDA  OF  ITEMS  FOR  ESTIMATES.* 

Excavation,  per  cubic  yard,  ascertain  the  quantity  of  earth  to  be 
excavated  for  foundation  walls,  drains,  fence  posts,  etc. 

Foundation  Walls,  per  cubic  foot,  1 28  feet  to  the  cord  of  ma- 
sonry. Find  the  number  of  cubic  feet  in  walls  and  footing  courses, 
deducting  all  openings  over  9 feet  in  width. 

Tile  Drains. — Calculate  the  number  of  lengths,  bends  and  junc- 
tions required,  add  cost  of  laying  and  connecting,  including  cover- 
ing in,  etc. 

Galvanized  Iron  and  Lead  Pipes. — State  the  length  and  diame- 
ters of  all  galvanized  iron  pipes  necessary,  and  the  weight  of  lead 
pipes  per  yard  lineal,  together  with  all  traps,  overflow  pipes,  and 
cocks  necessary. 


* Builder’s  Guide  and  Estimator’s  Price  Book,  p.  9. 


io8 


PRACTICAL  CARPENTRY. 


Water  Closets.— If  water  closets  are  to  be  provided  state  how  to 
to  be  fitted  up. 

Bath . — State  description  of  bath,  if  of  galvanized  iron  or  other 
metal,  including  fixing  in  frame  and  casing. 

Brick  Walls  per  foot  cube.  20  bricks  (of  8 ins.  x‘2^)  are  gener- 
ally allowed  to  a cubic  foot.  Find  the  number  of  cubic  feet  in  the 
walls,  division  walls  and  chimneys,  deduct  all  openings.  Measure 
all  chimneys  as  solid. 

Carpentry.  Under  this  head  commence  with  the  heavy  timbers, 
such  as  flooring  joists,  roofing  wall  plates,  lintels,  bond  timber, 
wood-bricks,  insertions  for  cornices,  projections  for  galleries,  stud- 
ding for -partitions,  furring  for  ceilings,  skirting,  groining,  etc. 
Framing  for  stables,  fencing  and  posts. 

If  the  building  is  a frame  or  a balloon,  find  the  number  of 
studs,  sills,  rafters  and  other  timbers  used,  also  the  number  of  feet 
of  rough  boarding,  siding,  shingling,  nails,  and  other  items  neces- 
sary to  complete  a wooden  structure. 

Joiners'  Work . — This  will  include  all  Hoofs,  doors,  windows, 
blinds,  shutters,  casings,  skirting,  and  fittings  of  every  description  in 
wood  work,  all  the  different  sized  doors,  windows  must  be  kept 
separate. 

Stove  Pipe  Rings . — State  number. 

Mantel  Pieces  and  Grates . — Number  mantel  pieces  and  grates, 
and  state  price,  provide  for  hearth  stones  and  fixings  all  complete. 

Closets.— State  quantity  of  shelving  required,  cloak  nails,  hooks, 
etc.  Calculate  for  plasterings -and  skirtings. 

Pantry.—  Describe  the  fitting  up  of  pantry,  whether  with  cup* 
boards  or  open  shelves,  and  state  if  with  sinks. 

Kitchen . — State  how  to  be  fitted  up  with  shelves,  pantry,  closets, 
etc. 

Bell  Hanging. — State  number  of  bells,  fixed  complete. 

Gas  Pipes.— State  number  of  lights  in  each  room,  etc. 

Staircases. — Describe  the  different  stairs  and  their  length,  width 
and  thickness;  state  the  kind  of  balusters  and  their  number,  also 
newels,  give  the  dimensions  of  hand  rail. 

Roof. — Describe  the  kind  of  roof,  whether  metal,  and  what  kind. 


PRACTICAL  CARPENTRY. 


IO9 

If  slate,  number  of  plies  of  felt  under  it,  gravel — do.  If  shingles, 
state  ^hat  kind,  etc. 

Gutters  and  Conductors . — State  the  width  of  gutters  and  how  to 
be  lined,  and  also  length  and  direction  of  gutters. 

Outside  Porches . — Provide  for  all  double  doors,  and  construction 
of  porches  as  described  in  specifications  as  well  as  all  steps  leading 
thereto. 

Fences. — State  the  different  kinds  of  fences  and  take  a price  at  per 
lineal  ft.,  including  gates  and  everything  necessary  to  complete  them. 

Of  course  there  will  be  many  other  items  than  these  described  in 
a specification,  but  sufficient  has  been  stated  to  enable  the  builder 
to  calculate,  as  to  the  value  of  the  building  he  contemplates  erect- 
ing, before  giving  in  a tender  for  the  work. 

It  is  understood  that  a plan  and  specification  will  be  provided 
for  the  use  of  the  estimator.  It  is  not  necessary  that  the  plan,  and 
elevation  when  given,  should  be  in  ink,  or  very  elaborate.  A 
pencil  drawing  will  often  be  all  that  will  be  required. 

After  the  quantities  are  taken  out  and  written  down  in  the  form 
herein  given,  and  the  prices  current  for  material  and  labor  in  each 
particular  added,  the  prices  being  for  the  completed  work,  as,  for 
instance,  the  price  of  a door  should  mean  the  cost  of  a door,  frame, 
casings,  achitraves,  lock,  hinges,  mouldings,  fixing  and  hanging 
complete,  including  the  painting.  By  adopting  this  system  the  esti- 
mator will  know  that  each  item  is  complete,  and  it  will  be  almost 
impossible  to  err  in  the  final  result.  When  all  the  items  are  written 
up,  and  everything  is  known  to  be  entered,  the  totals  should  be 
made  up,  and  20  per  cent,  added  to  cover  contingencies. 

It  is  a convenient  practice  to  those  unaccustomed  to  taking  out 
quantities,  to  note  down  on  the  plan  of  each  room  the  quantity 
of  plastering,  comice,  flooring,  wainscoting,  windows,  doors,  blinds, 
etc.,  contained  therein,  and  abstract  afterwards  the  different  items 
under  their  proper  headings. 

Sufficient  has  now  been  said  on  this  subject  to  enable  the 
workman  to  estimate  with  a tolerable  degree  of  accuracy. 

Nads.— The  following  shows  how  many  nails  are  required  for 
doing  certain  kinds  of  work : 


I IO 


PRACTICAL  CARPENTRY. 


To  nail  on  1000  shingles  from  $y2  to  5 lbs.  of  fourpenny  nails 
or  3 to  4 lbs.  of  threepenny  nails. 

1000  lath  requires  about  6]/2  lbs.  of  threepenny  nails. 

1000  feet  of  clapboards  requires  18  lbs.  of  sixpenny  box  nails. 


IOOO 

“ boarding  “ 

20 

it 

eightpenny  nails. 

IOOO 

it  it  it 

25 

it 

tenpenny  “ 

IOOO 

“ top  floors,  sq.  edge 

38 

a 

a it 

IOOO 

(t  it  tt 

41 

tt 

twelvepenny  “ 

IOOO 

“ matched  flooring 

31 

tt 

tenpenny  “ 

IOOO 

tt  tt  tt 

42 

tt 

twelvepenny  “ 

IOOO 

“ studding 

1 

tt 

tenpenny  “ 

IOOO 

“ furring,  1x3 

45 

tt 

tt  a 

IOOO 

“ “ 1x2 

6S 

a 

it  t( 

IOOO 

“ fine  finish 

30 

tt 

eightpenny  finish  nails. 

The  following  shows  how  many  nails 

to 

the  pound  there  are  in 

the  different  sizes : 


Name. 

Length. 

Number  to 

3d.  fine, 

i inch, 

557 

3d- 

1%  “ 

385 

4d. 

13/8  “ 

254 

Sd- 

*/4  to  1 inch. 

232  to  180 

6d.  finish, 

2 “ 

215 

6d. 

2 “ 

!54 

?d. 

“ 

141 

8d.  finish, 

2>4  “ 

*5° 

8d. 

2>4  “ 

90 

9d. 

2 3/  “ 

76 

iod.  finish, 

3 “ 

84 

iod. 

3 “ 

62 

1 2d. 

S/i  “ 

5° 

2od. 

3s/&  “ 

36 

3°d. 

4 “ 

24 

4od. 

4)4  “ 

18 

5°d. 

S/4  “ 

*3 

6od. 

6 “ 

9 

70& 

7 “ 

6 

PRACTICAL  CARPENTRY. 


Ill 


The  numbers  given  are  about  right,  but  sometimes  there  may  be 
more  or  less  per  pound  than  as  above,  as  some  manufacturers 
make  their  nails  lighter  in  weight  than  others.  In  estimating  for 
work,  however,  the  above  will  be  found  pretty  nearly  right. 

Cornices. — In  the  course  of  my  experience  I have  frequently 
been  asked  as  to  what  were  the  proper  proportions  for  a cornice  ? 
This  question  is  somewhat  difficult  to  answer,  on  account  of  the 
great  number  of  conditions  required  to  be  known  before  a correct 
one  can  be  given.  Of  course,  if  the  drawings  for  a building  are 
prepared  by  an  experienced  architect — which  they  ought  to  be 
where  possible— there  will  be  no  trouble  in  deciding  on  the  proper 
dimensions  for  projections  and  widths  of  the  various  members. 
For  country  carpenters,  however,  who  have  to  depend  on  their  own 
knowledge  of  proportion  for  deciding  on  the  various  sizes  and  di- 
mensions constituting  a cornice,  I give  the  following  hints,  though 
it  must  be  understood  that  the  sizes  here  given  may  be  varied  some- 
what to  suit  different  locations  and  different  tastes. 

For  dwellings  that  are  nicely  finished,  it  is  usual  to  allow  one  and 
a half  inches  projection  of  soffit  for  every  foot  in  height,  measuring 
from  sill  to  plate. 

If  the  buildings  are  more  than  two  and  a half  stories  in  height, 
this  projection  would  be  too  much,  and  where  the  buildings  are 
only  one  story  in  height,  the  proportion  would  be  too  little. 

For  verandas,  window  and  door  cornices,  piazzas,  porches  and 
bay-windows,  one  and  a half  inches  projection  to  each  foot  in 
height  will  be  about  right. 

The  width  of  the  frieze  will  depend  somewhat  on  its  position ; in 
a frame  house,  two  stories,  it  may  be  anywheres  from  twelve  to  six- 
teen inches  wide,  according  to  the  number  of  embellishments  or 
mouldings  it  contains.  The  plainer  it  is  the  narrower  it  may  be. 
The  taller  the  building  the  wider  the  frieze  will  require  to  be,  and 
buildings  less  than  two 'stories  in  height  should  not  have  friezes 
more  than  from  ten  to  twelve  inches  in  width. 

Crown  mouldings  or  facias  should  never  be  more  than  half  the 
width  of  the  frieze,  and  in  a majority  of  cases  should  be  less.  When 
they  are  moulded  and  have  several  members  planted  or  “stuck0  on 


X I 2 


PRACTICAL  CARPENTRY. 


them,  they  may  be  wider  than  when  plain.  In  the  Queen  Anne  style  oi 
building,  where  the  frieze  and  facia  are  loaded  with  small  mouldings, 
beads  and  rosette  ornaments,  a greater  width  of  surface  is  admissible. 

In  designing  verandas,  porches,  bay-windows  and  similar  work, 
a great  deal  of  latitude  may  be  allowed,  and  the  width  of  the  frieze 
may  be  varied  from  six  to  ten  inches,  or  in  the  case  of  arched  open- 
ings the  frieze  may  be  made  from  eight  to  eighteen  inches  in  width, 
to  suit  the  surrounding  conditions.  Of  course,  wide  friezes  or  facias 
on  verandas  must  be  relieved  with  mouldings,  or  be  broken  into 
panels  at  regular  intervals  or  at  symmetrical  distances;  or  the 
frieze  may  have  brackets,  singly  or  in  pairs  or  triplets,  placed  at 
regular  or  symmetrical  irregular  distances,  in  such  a manner  as  to  re- 
lieve the  monotony  of  the  wide  field  of  frieze  exposed.  The  crown 
or  upper  portion  of  the  cornice  above  the  projection,  which  is  also 
called  the  facia,  must  not  increase  in  width  on  verandas,  piazzas, 
porches  and  bay-windows,  in  the  same  proportion  as  the  frieze,  and 
the  designer  must  be  very  careful  when  preparing  drawings  or 
sketches  for  this  work,  and  avoid  long  flat  and  unbroken  surfaces 
on  facias.  On  no  part  of  the  exterior  of  a building  is  good,  honest 
work  more  necessary  than  on  the  cornices  over  bay-windows, 
porches  and  verandas;  every  mitre  and  joint  should  be  true  and 
perfect,  and  every  running  and  other  ornament  should  be  straight, 
regular  and  smooth. 

Base-boards  or  Plinth. — The  height  of  the  base-boards  of  a room 
is  governed  somewhat  by  the  height  of  the  room  itself,  and  to  a 
certain  extent  also  by  the  length  of  the  straight  walls.  One  and  an 
eighth  inches  to  every  foot  in  height  of  the  room  gives  a very  fair 
proportion,  but  in  most  cases  one  and  a quarter  inches  suits  much 
better;  in  rooms  of  great  length,  however,  where  the  walls  are 
straight  and  unbroken,  an  extra  inch  or  more  might  be  added  to 
the  whole  width,  which  will  have  a tendency  to  give  it  a more 
pleasing  effect  than  if  left  narrower.  In  many. cases  the  width  or 
height  of  a base  will  be  determined  by  circumstances  or  by  the 
taste  of  designer.  The  wider  the  base  the  more  mouldings  it 
should  have  upon  it,  but  the  mouldings  should  never  project  past 
the  face  of  the  base. 


PRACTICAL  CARPENTRY 


1 *3 

A base-board  may  be  square  or  plain,  ornamented  by  a bead  or 
moulding  stuck  upon  it,  or  by  a detached  moulding ; or  it  may  be  sunk 
to  form  a double  plinth,  with  or  without  mouldings  on  each  offset 

Sometimes  the  base-board  is  let  into  a groove  in  the  floor,  which 
prevents  the  crack  or  opening  that  always  takes  place  between  the 
floor  and ‘the  lower  edge  of  the  base  when  the  latter  shrinks. 

This  practice  is  not  so  common  in  America  as  in  England,  but 
as  an  excellent  substitute  we  adopt  the  following  method : Pro- 

cure a piece  of  stuff— hard  wood  ^is  the  best  for  the  purpose — 
about  three-quarters  of  an  inch  wider  than  the  base  is  thick,  and 
about  one  inch  thick.  A groove  is  then  made  in  the  piece  about  a 
half  an  inch  deep,  and  half  an  inch  wide,  and  three-quarters  of  an 
inch  from  the  face  of  the  stuff  showing  in  the  room.  A moulding 
may  be  stuck  on  the  face-edge,  or  it  may  be  rounded  off  from  the 
edge  of  the  groove  to  the  floor.  This  strip  is  carefully  nailed  down 
on  the  floor  in  proper  place  and  straight,  nicely  mitered  at  the  cor- 
ners in  both  re-entrant  and  external  angles,  and  left  level  and 
straight  on  the  top.  The  base-board  is  then  made  with  a lip  on 
its  lower  edge,  gauged  from  its  face  and  made  just  the  size  to  fill 
the  groove  made  in  the  running  piece.  This  lip  is  forced  into  the 
groove,  and  the  base-board  is  then  nailed  on  to  the  grounds,  stud- 
ding or  other  means  of  fastening.  This  makes  very  good  work, 
for  should  the  joists  in  the  floor  shrink  and  let  the  floor  down,  or 
the  base  shrink,  the  lip  would  draw  out  of  the  groove  a little,  but 
no  joint  would  be  visible,  and  no  wind  or  vapors  could  get  into 
the  room  from  this  source. 

If  no  provision  of  this  kind  is  made  to  meet  the  shrinkage,  then 
the  base-board  must  be  “ scribed  ” to  the  floor,  so  that  it  will  fit 
closely  at  every  part. 

It  is  best  to  dove-tail  the  corners  at  the  internal  angles,  and 
mitre  the  external  angles.  All  butt-joints  should  have  a tongue  or 
feather  in  them.  This  keeps  them  even  on  the  face  and  makes  the 
joint  so  much  stronger.  In  all  cases  of  corners,  the  mouldings  must 
show  a mitred  angle,  either  by  being  cut  so  or  by  being  scribed. 

Double  bases  consist  of  two  base-boards,  joined  together  in  the 
direction  of  their  width.  The  members  may  be  of  equal  width  or 


1 14  PRACTICAL  CARPENTRY. 

the  lower  part  may  be  narrower  or  wider,  according  as  the  design 
or  conditions  demand  it.  Wherever  the  joint  takes  place  there 
ought  to  be  a moulding,  bead  or  offset  stuck  on  the  stuff,  so  as  to 
cover  the  line  of  junction,  which,  in  case  of  shrinkage,  hides  the 
joint. 

Dado , Wainscot,  or  Surbase. — For  the  sake  of  ornament,  and  to 
prevent  the  wall  from  being  injured  by  chairs  knocking  up  against 
it,  a moulded  bar,  called  a “ chair-rail,”  is  sometimes  fixed  at  a 
height  of  about  three  feet  from  the  floor  and  parallel  with  the  base. 
The  rail  should  be  fastened  well  to  the  wall-furring  or  grounds  pro- 
vided for  that  purpose.  These  rails  should  never  be  less  than  three 
inches  wide.  The  interval  between  the  rail  and  base  is  called  the 
“ dado,”  and  the  whole  work,  base,  dado,  and  rail  combined,  is 
often  called  by  workmen  the  wainscot  or  wainscoting,  and  the  top 
rail  alone  is  termed  the  surbase.  Sometimes,  in  halls,  hotels, 
churches  and  public  buildings,  the  wainscoting  is  made  as  high  as 
six  feet  or  more  from  the  floor,  and  is  broken  into  panels  of  various 
forms  or  made  up  of  narrow  strips  matched  and  beaded  or  left  with 
a V joint.  Sometimes  a tier  of  ornamental  tiles  is  run  around  at 
an  appropriate  distance  from  the  surbase,  with  a second  rail  and 
moulding  running  under  the  tile  line. 

Woods. — In  these  days  every  carpenter  and  joiner  should  be 
able  to  manipulate  all  our  native  hard  woods,  as  the  day  is  at  hand 
when  nearly  all  the  better  sort  of  houses,  public  buildings,  churches, 
schools,  club-rooms,  stores,  banks  and  offices  will  be  finished  in 
hard  wood ; therefore,  it  is  quite  necessary  that  workmen  should’ 
become  acquainted  with  the  methods  of  working  and  finishing 
them,  and  my  advice  is  that  each  workman  that  desires  to  push 
himself  forward  in  his  trade,  should  acquire  a knowledge  of  the 
various  kinds  of  hard  woods,  their  qualities  and  adaptabilities  for 
various  purposes.  The  country  workman  scarcely  ever  works  any 
other  woods  than  pine  and  hemlock,  and  sometimes,  perhaps,  spruce 
and  a little  walnut.  He  has  no  idea  of  the  fine  effect  produced  by 
a combination  of  ash  and  walnut;  or  black  birch,  or  cherry  and 
maple,  or  of  oak  and  walnut,  when  properly  handled ; or  our  coun- 
try houses  would  contain  more  rooms  finished  off  with  native  hard 


PRACTICAL  CARPENTKi . 


**5 


woods  than  we  now  have.  A parlor  or  dining-room  finished  with 
some  of  our  common  native  hard  woods  costs  but  very  little  more 
than  the  same  room  would  finished  with  pine  and  daubed  over 
with  paint  or  some  other  abomination.  It  is  astonishing  what  a 
fine  effect  white  ash  has  when  used  as  finish  in  a country  house, 
when  properly  wrought  and  prepared.  The  best  filling  for  woods 
of  this  kind  is  the  now  celebrated  “ Wheeler’s  Wood -filler,1 ” which 
may  be  found  done  up  in  cans,  in  almost  any  paint-store  in  the 
country,  directions  for  using  accompanying  each  can.  Heretofore, 
one  of  the  great  objections  urged  against  the  use  of  hard  wood  for 
finish  by  country  workmen  was  the  difficulty  of  finishing  the  work 
after  it  had  been  put  in  place.  To  oil  or  varnish  the  wood  before 
filling  was  simply  to  spoil  the  whole  work  and  make  it  look  worse 
than  if  painted,  which,  indeed,  it  had  to  be  in  a great  many  cases, 
to  hide  its  ugliness.  A filler,  however,  properly  applied,  fills  up 
the  pores  of  the  wood  and  hardens  its  surface,  and  excludes  the 
dust  and  dirt  from  penetrating  into  the  wood  and  disfiguring  it,  and 
varnish  applied  on  this  shows  an  even  surface  that  is  at  once  rich 
and  pleasing. 

The  following  tables  refer  more  particularly  to  timbers  intended 
for  constructive  than  for  decorative  pur-  oses,  and  wili  be  found 
more  useful  to  the  carpenter  than  to  the  joiner,  but  in  many  cases 
will  be  found  useful  to  both. 

Mr.  Hodgkinson  found  that  timber  when  wet  had  not  half  the 
strength  of  the  same  timber  when  dry.  This  is  an  important  point 
to  consider  in  subaqueous  structures. 

Resistance  to  Crushing  Across  the  fibres. — When  a vertical  piece 
of  timber  stands  upon  a horizontal  piece,  the  latter  is  compressed 
at  right  angles  to  the  length  of  the  fibres,  and  in  this  position  it  will 
not  withstand  so  great  a compressive  force  per  square  inch  as  does 
the  vertical  piece,  whose  fibres  are  compressed  in  the  direction  of 
their  length. 

Not  many  experiments  have  been  made  on  this  point. 

Tredgold  found  that  Memel  pine  was  distinctly  indented  with  a 
pressure  of  1000  lbs.  per  square  inch,  and  English  oak  with  1400 
lbs.  per  square  inch. 


PRACTICAL  CARPENTRY 


n6 


TABLE  SHOWING  THE  WEIGHT,  STRENGTH,  ETC.,  OF  VABlOUS  WOODS. 


Wood  seasoned. 

Weight  of  a 
cubic  foot 
(dry). 

Tenacity  per 
Sq.  inch. 

lengthways 
of  the  gram 

Modulus 

of 

Rupture 

Modulus 

of 

Elasiicity 

Lbs. 

Tons. 

Lbs. 

Lbs. 

Acacia 

48 

From 

50 

: To 

8'i 

1,152,000  to 

Alder 

50 

4*5  . 

6'* 

1.687,500 

1,086,750 

Ash,  English 

43  to  53 

i-8 

7*6 

12,000  to 

1,525,500  to 

**  American 

30 

2*45 

14,000 

10,050 

2,290,000 

1,380.000 

Beech 

43  to  53 

2*1 

6-6 

9,000  to 

1,350,000 

Birch 

4 f to  49 

67 

1 2,OCO 
11.700 

1,645,000 

Cedar 

35  to  4 7 

i*3 

5*i 

7,400  to 

486,000 

Chestnut 

35  to  4t 

4*5 

5*8 

8,000 

10,660 

1,140,000 

Elm,  English 

34  to  37 

24 

6*3 

6,000  to 

700,000  to 

**  American 

47 

4’i 

9,700 

14,490 

1,340,000 

2,470,000 

Fir,  Spruce 

29  to  32 

i*3 

4*5 

9,900  t( 

1,400,000  to 

“ Dantzic 

36 

1*4 

4*5 

12,300 

13,806 

1,800,000 
2,  300,  COO 

41  American  . red 

34 

1*2 

60 

7,  roo  to 

1,460,000  to 

pine. 

44  American  ‘ yel- 

32 

°*9 

10,290 

8.454 

2.350.000 

1.600.000  to 

pine .... 

44  Memel 

34 

4*2 

4*9 

2,480,000 
I,  ; 36, 000  to 

44  Kaurie 

34 

2*0 

IL334 

14,088 

1,9*7*750 

2,880,000 

44  Pitch  pine. 

41  to  58 

2 ’X 

4*4 

1,252,000  to 

44  Riga. 

34  to  47 

1.8 

5-5 

9.450 

3,000,000 
870.000  to 

Greenheart 

58  t©  72 

3*9 

4*i 

16,500  to 

3,000.000 

1,700,000 

Jarrah 

Larch 

63 

i*3 

27.500 

10,800 

1,187,000 

32  to  38 

i*9 

5*3 

5,000  to 

1,360,000 

Mahogany,  Spanish. . 

1 S3 

*7 

7*3 

10,000 

7,600 

1,255,000  to 

44  Honduras 

35 

i*3 

84 

1 1,500  to 

3, 

1,596,000  to 

Mora 

57  to  68 

4*i 

12.600 
21,000  to 

1,970,000 
1,86c, 000 

Oak,  English 

49  to  58 

3*4 

8 8 

22,000 
10,000  to 

1 ,200,000  tO 

**  American  

6r 

3*° 

4*6 

13,600  • 
12,600 

1,750,000 

2,100,000 

Plane 

40 

5*4 

i»343>25° 

Poplar 

23  to  26 

2 68 

763,000 

Sycamore 

Teak 

36  to  43 

4*3 

58 

9.600 

1,040.000 

41  to  52 

1*47 

6*7 

• 2,000  tO 

2,167,000  to 

Willow 

24  to  35 

6*25 

19,000 

T5,6oO 

2,414,000 

Hornbeam 

475 

9 1 

** 

2 =<- 

o W);". 

SJ-gg 
•1  i ££ 


Tons  per 

>q. 

inch. 

>» 

V 

*5) 

iderai 

dry. 

0 

0 

s 

M 

H 

3*8 

4*2 

2*5 

3*4 

4*2 

i*5 

2-8 

2*5 

2 '6 

2 6 

4*6 

4*i 

2*9 

3° 

3’1 

2*1 


i-8 

6 

2*6 

3'o 

2*1 

5*8  6*8 

3‘2 

2*6 

3*2 

2*7 


2*9  4 ’5 

3*1 

»*4  2*3 

3 » 

2’3  5 ’4 

i 3 27 

3 7 


Comparative 
Stiffness  and 
Strength,  accord- 
ing to  J redgold. 
Oak  being  ioo.a 


Srift- 

ness. 

Strength. 

98 

95 

63 

80 

89 

119 

77 

79 

77 

103 

28 

62 

67 

89 

78 

82 

139 

114 

72 

86 

130 

108 

132 

81 

139 

66 

114 

80 

162 

89 

73 

8s 

62 

83 

98 

165 

67 

8.5 

79 

103 

73 

67 

93 

96 

105 

164 

100 

100 

114 

86 

78 

92 

44 

50 

82 

111 

126 

109 

ro8 


l From  Hodgkinson’s  experiments  on  short  pillars  i inch  diameter,  2 inches  high,  flat  ends, 
and  Laslett*'  on  2-inch  cubes. 

a This  ratio  is  not  always  confirmed  by  the  values  of  the  moduli  of  elasticity  as  found  by 
more  recent  experiments,  and  given  i • the  fif.h  colimn  of  ihe  ab  jve  table. 


PRACTICAL  CARPENTRY. 


117 


Mr.  Hatfield’s  experiments  chiefly  on  American  woods,  are 
quoted  in  Hurst’s  Tredgold and  form  the  basis  of  a table  in  Hurst’s 
Pocket-Book , from  which  the  following  are  taken  : — 

Force  per  sq.  inch  required  to 
crush  the  fibres  transversely 
1-20  inch  deep. 


Fir,  spruce,  - - - -22  tons. 

Pine,  Northern  or  Yellow,  - - *60  “ 

44  White  (P.  strobus ) American,  - - *27  “ 

Mahogany,  Honduras,  - - - - *58  44 

44  St.  Domingo,  - - - 1-92  44 

Oak,  English,  - - - - *90  44 

“ American,  -----  *84  44 

Ash,  American,  - - - - 103  44 

Chestnut,  - - - - - -42  44 


Resistance  to  Shearing . — -On  this  point  also  but  few  experiments 
have  been  made. 

The  resistance  to  shearing  in  direction  of  the  fibres  of  the  wood 
is  of  course  much  less  than  that  across  the  fibres. 


Wood. 

Resistance  to  Shearing 
per  sq.  inch  in  lbs. 

Along  Fibres. 

Across  Fibres. 

Fir 

556  to  634' 
2300  = 
7803 
1400  » 
600  » 

500  to  8OO2 

4000  > 

Oak 

American  oak 

Ask  and  elm 

Spruce 

Bed  pine 

1 Barlow  On  Strength  o f Materials , p.  23.  2Rankine’s  Civil  Engineering. 

3 Hatfield,  quoted  in  Hurst’s  Tredgold. 

If  the  reader  wishes  to  pursue  this  subject  further  he  will  find  it 
fully  discussed  in  Hatfields  dransverte  Strains , price  $6.00 ; 
Hurst's  Tredgold , price  $6.00;  The  Builder's  Guide  and  Esti- 
mator’s Price- Book,  price  $2.00  ; and  in  Gwilfs  Encyclopedia  oj 
Architecture,  price  $18.00. 

The  following  items  are  submitted,  as  they  place  before  the 
reader,  in  a handy  form,  a few  solutions  of  matters  that  will  fre- 
quently present  themselves  to  the  active  worker: 

Timber  for  Posts  is  rendered  almost  proof  against  rot  by  thorough 
seasoning,  charring,  and  immersion  in  hot  coal  tar. 


ii8 


PRACTICAL  CARPENTRY. 


Increase  in  Strength  of  Different  Woods  by  Seasoning. — White 
pine,  9 per  cent.;  elm,  12*3  per  cent.;  oak,  266  per  cent.;  ash, 
447  per  cent. ; beech,  61*9  per  cent. 

Comparative  Resilience  of  various  Kinds  of  Timber. — Asli  being  1, 
fir  *4,  elm  *54,  pitch  pine  *57,  teak  *59,  oak  -63,  spruce  *64,  yellow 
pine  *64,  cedar  -66,  chestnut  73,  larch  84,  beech  *86.  By  resili- 
ence is  meant  the  quality  of  springing  back,  or  toughness. 

To  Bend  Wood. — Wood  enclosed  in  a close  chamber,  and  sub 
mitted  to  the  action  of  steam  for  a limited  time,  will  be  rendered  so 
pliant  that  it  may  be  bent  m almost  any  direction.  The  same  process 
will  also  eliminate  the  sap  from  the  wood  and  promote  rapid  seasoning. 

Fireproofing  for  Wood. — Alum,  3 parts;  green  vitriol,  1 part; 
make  a strong  hot  solution  with  water;  make  another  weak  solu- 
tion with  green  vitriol  in  which  pipeclay  has  been  mixed  to  the  con- 
sistence of  a paint.  Apply  two  coats  of  the  first  dry,  and  then  fin- 
ish with  one  coat  of  the  last. 

To  Prevent  Wood  from  Cracking. — Place  the  wood  in  a bath  of 
fused  paraffin  heated  to  2120  Fahr.,  and  allow  it  to  remain  as 
long  as  bubbles  of  air  are  given  oft.  Then  allow  the  paraffin  to 
cool  down  to  its  point  of  congelation,  and  remove  the  wood  and 
wipe  off  the  adhering  wax.  Wood  treated  in  this  way  is  not  likely 
to  crack. 

Comparative  Value  of  Different  Woods7  showing  their  crushing 
strength  and  stiffness  : — Teak,  6,555;  English  oak,  4,074;  ash, 
3,571;  elm,  3,468;  beech,  3,079;  American  oak,  2,927;  mahog- 
any,  2?57 1 ; spruce,  2,522;  walnut,  2,374;  yellow  pine,  2,193; 
sycamore,  1,833;  cedar,  700. 

Relative  Hardness  of  Woods. — Taking  shell-bark  hickory  as  the 
highest  standard,  and  calling  that  100,  other  woods  will  compare 
with  it  for  hardness  as  follows  : — Shell-bark  hickory,  100;  pig-nut 
hickory,  96;  white  oak,  84;  white  ash,  77;  dogwood,  75;  scrub 
oak,  73;  white  hazel,  72;  apple  tree,  70;  red  oak,  69;  white 
beech,  65 ; black  walnut,  65 ; black  birch,  62 ; yellow  oak,  60  ; 
white  elm,  58;  hard  maple,  56  : red  cedar,  56;  wild  cherry,  55  ; 
yellow  pine,  54;  chestnut,  52;  yellow  poplar,  51;  butternut,  43; 
white  birch,  43  ; white  pine,  30. 


PRACTICAL  CAPvPENTRY. 


ll9 


Tensile  Strength  of  D iff eient  Kinds  of  Woods , showing  the  weight 
or  power  required  to  tear  asunder  one  square  inch : — Locust, 
25,000  lbs.;  mahogany,  21,000  lbs.;  box,  20,000  lbs.;  bay, 

14.500  lbs.;  teak,  14,000  lbs.;  cedar,  14,000  lbs. ; ash,  14,000  lbs. ; 
oak,  seasoned,  13,600  lbs.;  elm,  13,400  lbs.;  sycamore,  13,000 
lbs.;  willow,  13,000  lbs;  mahogany,  Spanish,  12,000  lbs.;  pitch 
pine,  12,000  lbs  ; white  pine,  11,800  lbs. ; white  oak,  11,500  lbs. ; 
lignum  vitae,  11,800  lbs.;  beech,  11,500  lbs.;  chestnut,  sweet, 

10.500  lbs.;  maple,  10,500  lbs.;  white  spruce,  10,290  lbs.;  pear, 
9,800  lbs.;  larch,  9,500  lbs.;  walnut,  7,800  lbs.;  poplar,  7,000 
lbs.;  cypress,  6,000  lbs. 

Cisterns. — The  capacity  of  a cistern  is  estimated  in  gallons;  7 y? 
gallons  to  a cubic  foot,  which,  though  not  strictly  correct,  is  near 
enough  for  practical  purposes ; hence,  to  get  the  contents  of  a cis- 
tern in  gallons,  multiply  the  product  of  the  length,  breadth,  and 
depth  of  the  inside,  in  feet,  by  7 ]/2. 

The  following  table  will  give  the  contents  or  capacities  of  cisterns 
from  two  to  twenty-five  feet  in  diameter,  with  a depth  of  ten  inches. 
The  number  of  gallons  may  be  multiplied  by  the  number  of  times 
ten  inches  will  divide  in  the  depth  of  the  cistern  ; thus  r I make  a 
cistern  forty-five  inches  deep  and  five  feet  in  diameter,  then  I mul- 
tiply 122*40  X 4*5  = 550*80  gallons,  or  nearly  551  gallons. 


2 feet 

24  44 

Gallons. 

19-5 

30-6 

3 44  

44  06 

3*  44 

59*97 

4 44  

78-33 

44  44  

9914 

5*  44 

122-40 

6*  •• 

14810 

6 44  

176-25 

6}  *' 

206-85 

7 44  

239-88 

7J  44  

275-40 

8 feet 

84  44  

Gallons. 

313-33 

353-72 

9"  44  

396-56 

94  44  

461-40 

10  44  

489-20 

11  44  

592-40 

12  44  

705-00 

13  44  

827*4 

14  44  

959-6 

15  44  

1101-6 

20  44  

1958-4 

25  44  

3059-9 

Stairs. — As  this  is  a subject  of  considerable  magnitude  it  is  not 
intended  to  enter  into  it  here  any  further  than  to  state  that  the 
author  of  the  present  work  has  in  preparation  a thorough  treatise 
on  the  subject,  which  will  cover  the  whole  ground  of  staircasings, 
bodies  and  handrailing,  and  it  is  intended  that  the  work  shall  be 


120 


PRACTICAL  CARPENTRY. 


issued  uniform  with  this  work  on  carpentry,  and  sold  at  the  same 
price ; namely — one  dollar. 

There  are  a great  many  works  on  stairbuilding  in  the  market,  all 
of  which  possess  more  or  less  excellence,  but  none  of  which  -seem  to 
/each  the  popular  requirement,  as  they  are  written  much  above  the 
heads  of  ordinary  workmen,  or  they  deal  altogether  with  hand- 
railing,  to  the  total  exclusion  of  the  carcass  and  body  of  the  stair. 
In  the  forthcoming  work  an  attempt  will  be  made  to  remedy  these 
defects  by  giving  special  attention  to  the  carcass  as  well  as  the  rail, 
and  by  putting  everything  connected  with  stairs  in  the  plainest 
phraseology,  and  so  that  everyone  may  understand,  whether  he  has 
a knowledge  of  geometry  or  not.  The  following  table  may 
be  of  use  here,  as  it  shows  the  relative  proportions  of  treads  and 
risers  to  each  other,  where  it  is  possible  to  follow  the  figures  here 
laid  down.  It  sometimes  happens,  however,  that  both  space  and 
height  will  determine  the  width  of  tread  and  height  of  riser.  When 
this  occurs,  the  workman  should  adhere  as  closely  as  possible  to  the 
proportions  herein  given : 


Width  of  Tread. 

6 inches . . 

7 “ -- 

8 “ . . 

9 “ -- 

10  “ 

11  “ 

12  “ . . 

13  “ 


Height  of  Riser. 

. 8}4  inches. 

8 “ 

lA  “ 

7 

■ ^A  “ 

. 6 “ 

sA  “ 


The  width  of  the  tread  does  not  include  the  “ nosing  ” or  pro* 
jection.  This  will  make  a 12  inch  tread  about  13!  inches  wide,  as 
the  step  or  tread  will  project  i|  inches,  or  at  least,  the  thickness  of 
the  stuff.  This  projection  never  counts  in  the  run  as  the  projection 
of  each  tread  stands  over  the  tread  below.  The  height  of  riser  in- 
cludes the  thickness  of  the  tread,  so  that  the  first  riser  requires  to 
be  narrower  by  the  thickness  of  the  tread  than  any  of  the  risers 
placed  above  it.  There  is  always  one  more  riser  than  treads  in 
every  flight  of  stairs.  This  is  owing  to  the  fact  that  the  floor  at 


PRACTICAL  CARPENTRY. 


121 


the  foot  of  the  stairs,  and  the  floor  at  the  landing,  take  the  places 
of  treads,  though  not  counted  as  such.  The  rules  and  hints  given 
in  the  foregoing  apply  as  well  to  winding  as  to  straight  stairs. 

Winders  are  measured  at  their  centres. 

To  estimate  the  cost  of  ordinary  stairs,  either  open  or  cased,  first 
calculate  the  number  of  feet  per  straight  step,  counting  the  step  and 
riser  and  the  strings  adjacent,  allowing  generally  i foot  in  length  of 
string  for  each  riser.  Allow  for  timbering  underneath  such  a part 
of  the  actual  number  of  feet  as  may  be  required  to  represent  its 
value  when  figured  at  the  same  price  as  other  lumber.  After  ascer- 
taining the  number  of  feet  in  one  straight  step,  as  above,  multiply 
by  the  whole  number  of  risers  in  the  stairs,  allowing  three  for  each 
winder  or  swelled  step  when  the  stairs  are  winding.  An  allowance 
of  two  will  be  enough  when  there  is  no  furring  underneath.  Allow 
four  for  quarter  platforms  and  six  for  half  platforms.  No  allowance 
need  be  made  usually  for  landings  unless  large.  Double  the  cost 
of  dressed  lumber.  Figure  pine  at  5 cents  per  foot.  Figure  oak 
at  6 cents  per  foot.  Figure  walnut  at  10  cents  per  foot,  costing  6 
cents.  For  ornamental  brackets  of  pine  allow  15  cents  each,  or 
per  foot  or  fascia.  For  oak  or  walnut  allow  25  cents  each,  or  per 
foot  of  facia.  For  plain  railing  put  up,  multiply  number  of  square 
inches  on  cross  section  by  4 cents,  which  will  be  the  price  per  foot. 
Figure  crooks  and  ramps  at  three  times  their  length  ; for  small  rails 
or  toad-back  rails  add  12  per  cent. 

Figure  ordinary  turned  ball  listers,  smoothed  and  dovetailed, 
2-inch  walnut,  at  15  cents;  2^-inch  walnut,  at  18  cents;  2^-inch 
walnut,  at  22  cents;  2-inch  oak,  at  10  cents;  2^-inch  oak,  at  15 
cents;  and  2j-inch  oak,  at  18  cents. 

Figure  plain  turned  newels,  walnut,  42  cents  per  inch  of  diameter 
(bases  dressed);  oak  at  35  cents  per  inch  of  diameter  (bases 
dressed);  octagon  newels  (bases  dressed)  walnut  at  64  cents  per 
inch  of  diameter,  and  oak  at  55  cents  per  inch  diameter;  octagon 
panelled  shaft,  walnut,  85  cents  per  inch  of  diameter,  and  oak  75 
cents  per  inch  of  diameter. 

Special  designs  may  be  figured  somewhat  in  the  same  manner, 
varying,  however,  as  the  design  seems  to  require. 


122 


PRACTICAL  CARPENTRY. 


These  prices,  of  course,  are  only  approximately  correct;  but 
when  reliable  figures  are  absent,  they  will  be  found  to  answer 
the  purpose  with  tolerable  accuracy. 

INCLINATIONS  OF  ROOFS. 

The  various  kinds  of  coverings  used  on  roofs  are  copper,  lead, 
iron,  tin,  slates,  tiles,  shingles,  gravel,  felt,  pitch  and  cement.  If  the 
inclinatidn  for  slates  is  26^°,  which  is  about  the  correct  thing,  the 
following  table  will  show  the  proper  degrees  of  inclination  suitable 
for  other  materials. 


Kind  of  Covering. 

Inclination  to  the  Hor- 
izon in  Degrees 

Height  of  roof 
in  parts  of  a 
Span. 

Weight  upon  a square 
of  roofing.* 

Deg, 

Min. 

Tin 

3 

50 

1-48 

50  pounds. 

Copper 

3 

50 

1-48 

100 

Lead 

3 

50 

1-48 

700 

Slates,  large 

22 

00 

1-5 

1120 

41  ordinary 

26 

33 

1-4 

900 

44  tine 

26 

33 

1-4 

500 

Plain  Tiles. 

Gravel 

Felt  and  Cement  

29 

41 

2-7 

1780 

Roofs  covered  with  felt,  cement  and  gravel  may  have  any  incli- 
nation on  which  the  gravel  will  stay  during  heavy  rains.  The  best 
incline  for  a gravel  roof  in  the  latitude  of  New  York  is  about  54  of 
an  inch  to  the  foot;  but  for  the  South  and  Southwest,  the  inclina- 
tion might  be  something  less.  Where  bricks  and  cement  are  used 
for  a roof  covering,  the  surface  might  be  “cambered  ” like  a ship's 
deck,  and  then  drained  from  both  eaves ; or  the  roof  might  have  a 
slight  inclination  in  one  or  more  directions. 

CONTENTS  OF  BOXES,  BINS  AND  BARRELS. 

For  Winchester  bushel,  contents  2,150*42  cubic  inches. 

A box  9 inches x 9 inches  X 6£  inches  deep  will  contain  1 peck. 

A box  12  inches  x 12  inches  X 7i  inches  deep  will  contain  J 
bushel. 

A box  14  inches  X *4  inches  X 11  inches  deep  will  contain  1 
bushel. 


♦Timber  aad  boards  included. 


PRACTICAL  CARPENTRY. 


12  3 

5-bushel  box  or  bin  : 30  inches  X 30  inches  X 12  inches  deep, 
or  25  inchcsX  25  inches  X 17  3-16  inches  deep. 

10-bushel  bin:  30  inches  X 3°  inches  X 24  inches  deep,  or  2 
feet  X 3V2  feet  X 21  5-16  inches  deep,  or  3^  feet  X 3l/i  feetX 
12  3-16  inches  deep. 

15-bushel  bin  : 3^  feet  X 3%  feet  X i8*^  inches  deep,  or  3 
feet  X 4 feet  X i8^4  inches  deep. 

20-bushel  bin:  3^  feet  X 3\  feet  X 24^4  inches  deep,  or  3! 
feet  X 4 feet  X 21  5-16  inches  deep,  or  3 feet  X 4 feet  X 24^ 
inches  deep. 

25-bushel  bin:  3 feet  X4  feetX  3llA  inches  deep,  or  3!  feetX 

feet  X 2 3 11-16  inches  deep,  or  3 feet  X 5 feet  X 24^3  inches 
deep. 

30-bushel  bin  : 3^  feet  X 4j  feet  X 28 1 inches  deep,  or  3 feet  X $ 
feet  X 29^  inches  deep. 

40-bushel  bin:  4 feet  X 5 feet  X 29^  inches  deep,  or  4 feetX 6 
feet  X 24^$  inches  deep. 

50-bushel  bin:  4 feet  X6  feetX 31^6  inches  deep,  or  4!  feetX  7 
feet  X23  lI' 1 6 inches  deep,  or  5 feetX 6 feet  X 24^3  inches  deep. 

A common  flour  barrel  will  hold  about  334  bushels  of  grain  or 
other  fine  stuff. 

For  coarse  stuff,  such  as  turnips,  potatoes,  apples,  etc.,  heaped 
measures  are  allowed  for.  Cubic  contents  of  bushel  2,7477  inches. 

5-bushel  box  or  bin  : 30  inches  X 30  inches  X ^S/i  inches  deep, 
or  2 feet  X 3 feet  X i$3A  inches  deep. 

10-bushel  bin:  2|feetX3|  feet X2l%  inches  deep,  or 3 feet X 4 
feetX  16  inches  deep. 

15-bushel  bin:  3 feetX 4 feet  X23^g  inches  deep. 

20-bushel  bin  : 3 feet X 4 feet  X 32  inches  deep,  or  3!  feetX 4 
feet  X 27^  inches  deep. 

25-bushel  bin:  3!  feetX 4 feet  X 34  inches  deep,  or  3 feet  X 5 
feetX 31^  inches  deep,  or  3!  feet  X 5 feetX  27 ^ inches  deep. 

30-bushe!  bin:  3 feet  X 5 feet  X 38  inches  deep,  or  3!  feetX  5 
feet  X 32^  inches  deep. 

40-bushel  bin  : 3*  feet  X 6 feet  X 36#  inches  deep,  or  4 feet  X 6 
feetX  31^  inches  deep. 


124 


PRACTICAL  CARPENTRY. 


50-bushel  bin:  4 feet  X 6 feet X 39^  inches  deep,  or  5 feet X 5 
feetX  38^  inches  deep,  or  5 feet  X 6 feet  X 31^  inches  deep. 

A common  flour  barrel  will  hold  about  2\  bushels. 

ARITHMETICAL  SIGNS. 

= Sign  of  equality,  or  equal  to. 

+ “ Addition,  plus,  or  more. 

— “ Subtraction,  minus,  or  less. 

X “ Multiplication. 

-r  “ Division. 

: Is  to.  } 

: : So  is.  > Signs  of  Proportion. 

: To.  ) 

•J  Square  root;  when  placed  before  a number,  the  square  root  is 
to  be  extracted,  as,  *J6 4 = 8. 

3y  Cube  root;  and  signifies  that  the  cube,  or  third  root  is  to  be 
extracted,  as  3<v/  64  = 4. 

A number  is  said  to  be  squared  when  it  is  multiplied  by  itself. 
To  cube  a number,  is  to  multiply  it  three  times  by  itself,  as  the  cube 
of  4 is  4X4X4  = 64. 

0 Degrees,  'minutes,  "seconds. 

In  Duodecimals,  ' denotes  primes , or  twelfths;  " seconds , or 
twelfths  of  primes ; thirds , or  twelfths  of  seconds ; thus,  the  term 

4 6'  3''  reads,  4 ft.  6 in.  and  3 twelfths. 

Decimal  point,  as,  *5  = five  tenths,  *05  = five  hundredths, 
2-8  = 2 and  eight  tenths. 

MENSURATION  OF  SUPERFICIES. 

To  Find  the  Area  of  a Square . 

Rule, — Multiply  the  side  by  itself,  or,  in  other  words,  the  base  by 
the  perpendicular. 

Example . — To  find  the  area  of  a square  whose  side  is  17  feet. 
17X17  = 289,  the  area  of  the  square  in  feet. 

To  find  the  side  of  a square,  the  area  being  given,  extract  the 
square  root  of  the  area. 


PRACTICAL  CARPENTRY. 


*2S 


To  Find  the  Area  of  a Rectangle . 

Rule.  — Multiply  the  length  by  the  breadth,  and  the  product  will 
^be  the  area. 

Example, — To  find  the  area  of  the  rectangle. 

ft.  in 

107  its  length. 

7*3  breadth. 

741 

27‘9 

Feet,  76-8*9 

To  Find  the  Area  of  a Rhombus  or  Rhomboides . 

Rule , — Multiply  the  base  by  the  perpendicular  height  and  half 
the  product  will  be  the  area. 

Multiply  the  length  by  the  perpendicular  breadth,  and  the  pro- 
duct will  be  the  area. 

Let  the  side  be  17  feet,  and  the  perpendicular  15  feet,  then 
1 7 X 15  = 255,  the  area  required. 

To  Find  the  Area  of  a Triangle . 

Rule, — Multiply  the  base  by  the  perpendicular  height,  and  half 
the  product  will  be  the  area.  Let  the  base  of  the  triangle  be  14 
feet  and  the  perpendicular  height  9 feet,  then 

14X9  = 126  -+  2 =63  feet  the  area  of  the  triangle. 

Another  Rule, — Add  the  three  sides  together,  and  from  half  the 
sum  subtract  each  side  separately ; then  multiply  the  half  sum  and 
the  three  sides  together,  and  the  square  root  of  the  product  will  be 
the  area  required. 

Let  the  sides  of  a triangle  be  30,  40,  and  50  ft.  respectively. 

30+40+50  t 20 

2 60,  half  the  sum  of  the  sides. 

60 — 50  = 10,  first  remainder. 

60  — 40  = 20,  second  remainder. 


126 


PRACTICAL  CARPENTRY. 


6o  — 30  =30,  third  remainder. 

Then  60  X 10  X 20,  X 30  = 360,000. 

And  the  square  root  of  360,000  is  equal  to  600,  the  area  in  ft. 

Any  Two  Sides  of  a Right-Angled  Triangle  being  given,  to  Find 
the  Third  Side. 

1.  When  the  base  and  perpendicular  are  given. 

Rule. — To  the  square  of  the  base  add  the  square  of  the  perpen- 
dicular, and  the  square  root  of  the  sum  will  give  the  hypothenuse. 

Let  the  base  of  the  right-angled  triangle  be  24,  and  the  perpen- 
dicular 18,  tVfin<J  the  hypothenuse  or  third  side. 

576  square  of  the  base. 

324  square  of  the  perpendicular. 

576  + 324  = 900. 

And  the  square  root  of  900  is  equal  to  30  feet,  the  length  of  the 
third  side. 

2.  When  the  hypothenuse  and  one  side  is  given. 

Rule. — Multiply  the  sum  of  the  hypothenuse  and  one  side  by 
their  difference  ; the  square  root  of  the  product  will  give  the  other 
side. 

If  the  hypothenuse  of  a right-angled  triangle  be  30,  and  the  prer 
pendicular  18,  what  will  be  the  base? 

30  + '18  = 48  sum  of  the  two  sides. 

30 — 18  = 12  difference  of  the  two  sides. 

48X  12  = 576. 

To  Find  the  Area  of  a Trapezium. 

Rule. — Divide  the  trapezium  into  'two  triangles  by  a diagonal 
drawn  from  one  angle  of  the  figure  to  another.  The  areas  of  the 
triangles  may  be  found  hy  the  rules  already  given,  and  the  sums 
will  give  the  area  of  the  trapezium.  It  is  unnecessary  to  give  an 
example  of  this  problem,  as  it  would  only  be  a repetition  of  what 
has  been  already  given,. 

Irregular  Polygons , or  Many-sided  Figures . 

It  is  only  necessary  to  reduce  them  into  triangles  and  parallelo 


PRACTICAL  CARPENTRY. 


T27 


grams,  and,  calculating  these  severally,  to  add  them  together ; the 
sum  will  give  the  area  of  the  figure. 

In  this  manner  the  land-surveyor  estimates  the  quantity  of  acres, 
roods  and  perches  contained  within  certain  boundaries,  and  it  may 
be  done  with  considerable  accuracy  by  subdividing  the  space  mU;l 
the  whole  area  is  contained  within  a number  of  single  figures.  The 
carpenter  and  joiner,  however,  has  seldom  a necessity  for  this 
mode  of  proceeding,  for  it  is  customary,  in  all  those  cases  where  a 
surface  has  a variable  height,  to  take  the  medium  between  the  two 
extremes,  and  consider  the  superficies  as  a parallelogram.  But,  as 
the  builder  is  sometimes  required  by  circumstances  to  measure  the 
ground  which  is  chosen  as  the  site  of  a building,  it  is  necessary  that 
he  should  be  able  to  do  so  when  required. 

To  Find  the  Dia?neter  or  Circumference  of  a Circle  the  Diameter 
or  CircumfereJice  being  Given . 

1.  To  find  the  circumference,  the  diameter  being  given. 

Rule. — As  7 is  to  22,  so  is  the  diameter  to  the  circumference. 

Example. — If  the  diameter  of  a circle  be  84-5  inches,  what  is  the 

circumference. 

As  7 is  to  22*o,  so  is  84*5  to  265,751  the  circumference  required. 

2.  To  find  the  diameter,  the  circumference  being  given. 

Rule. — As  22  is  to  7,  so  is  the  circumference  to  the  diameter. 

To  Find  the  Area  of  a Circle . 

1.  When  the  diameter  and  circumference  are  both  given. 

Rule. — Multiply  half  the  circumference  by  half  the  diameter,  and 
the  product  will  be  the  area. 

2.  When  the  diameter  is  given. 

Ride. — Multiply  the  square  of  the  diameter  by  7854,  and  the 
product  will  be  the  area,  or  the  diameter  by  the  circumference  and 
divide  by  4. 

3.  When  the  circumference  is  given. 

Rule. — Multiply  the  square  of  the  circumference  by  ’07958,  and 
the  product  will  be  the  area. 


128 


PRACTICAL  CARPENTRY. 


To  Find  the  Area  of  a Sector  of  a Circle . 

Rule . — Multiply  the  radius  of  the  circle  by  one* half  of  the  arc  of 
the  sector. 

To  Find  the  Area  of  the  Segment  of  a Circle . 

Rule. — Find  the  area  of  a circle  having  the  same  arc,  and  deduct 
the  triangle  formed  between  the  two  radii  and  the  chord  of  the  arc. 

Properties  of  the  Circle . 

Diameter  X 3*14159  = circumference. 

Diameter  X *8862  = side  of  an  equal  square. 

Diameter  X *7071  = side  of  an  inscribed  square, 

Radius  squared  x *314159  = area  of  circle. 

Radius  X 6*28318  = circumference. 

Circumference  -7-3*14159  = diameter. 

MEASUREMENTS  OF  SOLIDS. 

To  Find  the  Solidity  of  a Cube . 

A cube  is  a solid  enclosed  by  six  equal  square  surfaces. 

Ride . — Multiply  the  side  of  the  square  by  itself  and  that  product 
by  the  side  of  the  square. 

Example. — The  side  being  9 feet. 

9X9  = 81,  then 

81  X 9 = 729  = the  solidity  required. 

To  Find  the  Solidity  of  a Parallelopipedon . 

A parallelopipedon  is  a solid  having  six  sides.  Every  opposite 
two  being  equal  and  parallel  to  each  other. 

Ride. — Multiply  the  length  by  the  breadth,  and  the  product  by 
the  depth,  and  it  will  give  the  solidity  required. 

Example. — Length  82  inches,  breadth  54,  depth  10  inches. 

82  X 54  = 4428,  then 
4428  X 10  =44280,  the  solidity  required. 


PRACTICAL  CARPENTRY. 


I29 


To  Find  the  Solidity  of  a Prism . 

A prism  is  a solid,  the  ends  of  which  are  parallel,  equal,  and  of 
the  same  figure.  Specific  names  are  given  to  them,  according  to 
the  form  of  their  bases  or  ends. 

Ride . — Multiply  the  area  of  the  base  by  the  perpendicular  height, 
and  the  product  will  be  the  solidity  required. 

To  find  the  solidity  of  a rectangular  prism  whose  base  is  30 
inches,  and  height  53. 

30  X 53  = 1 590,  lthe  solidity  in  inches. 


To  Find  the  Solidity  of  a Cylinder . 

A cylinder  is  a round  prism,  having  circles  for  its  ends,  and  is 
formed  by  the  revolution  of  a right  line  about  the  circumference  of 
two  equal  circles  parallel  to  each  other. 

Rule.—  Multiply  the  area  of  the  base  by  the  perpendicular  heigh/ 
of  the  cylinder,  and  it  will  give  the  solidity. 

To  Find  the  Solidity  of  a Sphere . 

A sphere  is  a solid  formed  by  thejevolution  of  a semicircle  round 
h fixed  diameter. 

Rule. — Multiply  the  cube  of  the  diameter  by  *5236,  and  the  pro* 
duct  will  be  the  solidity. 

For  the  Area  of  a Sphere . 

Multiply  the  square  of  the  diameter  by  3-1416. 

For  the  Circumference. 

Multiply  the  diameter  by  3*1416. 

The  surface  of  a spherical  segment  or  zone  may  be  found  by  mul- 
tiplying the  diameter  by  the  height,  and  then  by  3*1416. 

The  solidity  of  a spherical  segment  or  zone  may  be  found  thus — 
to  3 times  the  square  of  the  radius  (or  half  of  the  diameter)  add  the 
square  of  the  height,  then  multiply  the  sum  by  the  height  and  the 
product  by  *5236. 


1 3° 


PRACTICAL  CARPENTRY. 


REGULAR  POLYGONS. 


Number  of 
Sides. 

Name. 

Area  when  dinni 
of  inscribed  circle 

Area  when  side 

Length  of  side 
when  perpendicular 

Perpendicular 
| when  side 

Radius  of  circum- 
scribed circle 
when  side 

Length  of  side, 
when  radius  of  cir- 
cumscribed circle 
= x. 

3 

Triangle 

1*299 

0*433 

3*464 

0*289 

*577 

1*732 

4 

Square 

1*000 

1*000 

2*000 

0*500 

*707 

1*414 

5 

Pentagon 

•908 

1*720 

1*453 

0*688 

*851 

1*176 

6 

Hexagon  

*866 

2*508 

1*555 

0*866 

1*000 

1*000 

7 

Heptagon 

*843 

3*634 

*963 

1*039 

1*152 

•868 

8 

Octagon 

*828 

4*828 

*828 

1*207 

1*307 

*765 

9 

Nonagon 

*819 

6*182 

*728 

1*374 

1*462 

*684 

10 

Decagon  . . { 

*812 

7*694 

*650 

1*539 

1*618 

*618 

11 

Undecagon 

*807 

9*366 

*587 

Area  of  Polygons . 

Rule. — Multiply  the  square  of  the  side  by  the  figures  in  column  2* 


Trigon  3 sides  0*4330 

Pentagon 5 sides  1*7203 

Hexagon 6 sides  2*5981 

Heptagon 7 sides  3*6339 

Octagon 8 sides  4*8284 

Nonagon 9 sides  6*1818 

Decagon 10  sides  7*6942 

Undecagon 11  sides  9*3656 

Dodecagon 12  sides  11*1962 


Surfaces  and  Solidities  of  Regular  Bodies . 

Rule . — For  the  “surface”  multiply  the  square  of  the  length  of 
one  of  the  edges  by  column  2,  and  for  the  solidity  multiply  the  cube 
of  the  length  by  column  3. 

Surface.  Solidity. 


Tetraedron 4 faces  1 *7321  0*1178 

Hexaedron 6 " 6*0000  1*0000 

Octaedron 8 44  3*4641  0*4714 

Dodecaedron 12  44  20*6458  7*6631 

Icasaedron 20  44  8*6603  2*1817 


Number  for  Calculating  Areas . 

Circum.  of  circle  = Diam.  X 3*1416,  or  by  3 i-7th. 

Length  of  arc  of  circle  = Take  span  from  8 times  the  chord  of 
half  the  arc  and  one-third  remainder  = length  of  arc  required. 


PRACTICAL  CARPENTRY. 


l.V 


Ditto  when  arc  contains  120°=  Span  X 1*2092. 

Area  of  circle  = Square  of  diam.  X 7854. 

Area  of  segment  of  circle  = To  twice  square  root  of  span  plus 
square  of  rise  add  chord  of  half  arc,  the  result  multiplied  by  4-1 5 of 
rise  equals  area. 

Area  when  it  contains  120°  = Square  of  spanX  *20473. 

Area  of  sector  of  circle  = Radius  X half  the  length  of  arc. 

Area  of  ellipse  = Product  of  the  two  diameters  X *7834. 

Area  egg-shaped  sewer  =Square  of  transverse  diameterX  1*1597. 

Solidity  of  a cone  = Area  of  base  X one-third  perpendicular 
height. 

Solidity  of  globe  = Cube  of  diameter  X *5236. 

Prismoidal  formula  = Sum  of  end  areas  plus  4 times  middle  area 
multiplied  by  one-sixth  of  length. 

Gunted s Chain . 

Generally  adopted  in  land  surveying,  is  22  yards  in  length,  or 
100  links  of  7*92  inches  long.  The  length  was  fixed  at  22  yards 
because  the  square  whose  side  is  22  contains  exactly  i-ioth  of  an 
acre— or  1 chain  in  width  and  10  in  length  contains  an  acre;  80 
chains  make  1 mile,  and  a square  mile  is  the  square  of  80,  or  640 
acres. 


Diameters. 

TABLE  OF  SPHERICAL  CONTENTS. 
Surfaces. 

Capacities. 

1 

3141 

•523 

2 

12)567 

4-188 

3 

28-274 

14137 

4 

50-265 

33-51 

5 

78-540 

65-45 

10 

314159 

523-6 

15 

706-9 

1767  1 

20 

1256'6 

4189- 

25 

1963-5 

8181- 

30 

2827- 

14137- 

40 

5026- 

33510* 

DBAWINGS  AND  DRAWINo  INSTRUMENTS. 

Every  carpenter  should  know  how  to  measure  drawings,  and  the 
more  enterprising  ones  will  not  even  rest  satisfied  with  this,  but  will 

Note. — -The  prismoidal  formula  applies  to  earthworks,  casks,  and  truncate^ 
oonea. 


1 32 


PRAC1TCAL  CARPENTRY. 


endeavor  to  obtain  a knowledge  of  the  use  of  drawing  instruments 
and  drafting  generally.  To  aid  him  in  this  matter,  I have  thought 
it  proper  to  embody  the  following  hints  and  rules  on  the  subject  in 
the  present  work;  but  1 would  advise  the  ambitious  workman  to  se- 
cure one  of  the. standard  works  on  “Drawing  Instruments”  of 
which  there  are  many  to  be  found  in  the  book  stores.  Of  course, 
where  access  can  be  had  to  an  office  where  drafting  is  done,  the 
student  should  visit  there  at  once,  as  one  hour’s  experience  over 
a drawing-board  is  worth  a week’s  study  over  a book.  It  does  not 
take  a very  long  while  for  a good  workman  to  become  a fair  drafts- 
man when  once  he  settles  down  to  work. 

In  constructing  preparatory  pencil-drawings,  it  is  advisable,  as  a 
rule  of  general  application,  to  make  no  more  lines  upon  the  paper 
than  are  necessary ; and  to  make  them  as  dark  as  is  consistent  with 
the  distinctness  of  the  work.  And  here  I may  remark  the  inconven- 
ience of  that  arbitrary  rule  by  which  it  is,  by  some,  insisted  that 
the  pupils  should  lay  down  in  pencil  every  line  that  is  to  be 
drawn,  before  finishing  it  in  ink.  It  is  often  beneficial  to  ink  in  one 
part  of  a drawing,  before  touching  other  parts  at  all;  it  prevents 
confusion,  makes  the  first  part  of  easy  reference,  and  allows  of  its 
being  better  done,  as  the  surface  of  the  paper  inevitably  contracts 
dust,  and  becomes  otherwise  soiled  in  the  course  of  time,  and  there- 
fore the  sooner  It  is  done  with  the  better. 

Circles  and  circular  arcs  should,  in  general,  be  inked  in  before 
straight  lines,  as  the  latter  may  be  more  readily  drawn  to  join  the 
former  than  the  former  the  latter.  When  a number  of  circles  are  to 
be  described  from  one  centre,  the  smaller  should  be  drawn  and 
inked  first,  while  the  centre  is  in  better  condition.  When  a centre 
is  required  to  bear  some  fatigue  it  should  be  protected  with  a thick- 
ness of  stout  card,  glued  or  pasted  over  it,  to  receive  the  compass 
leg,  or  a piece  of  transparent  horn  should  be  used,  or  some  other 
suitable  material. 

India-rubber  is  the  ordinary  medium  for  cleaning  a drawing,  and 
for  correcting  errors  made  in  penciling.  For  slight  work  it  is  quite 
suitable;  but  its  repeated  application  raises  the  surface  of  the  paper, 
and  imparts  a greasiness  to  it,  which  spoils  it  for  line  drawing,  espe- 


PRACTICAL  CARPENTRY. 


!33 


dally  if  ink  shading  or  coloring  is  to  be  applied.  It  is  much  better 
to  leave  trifling  errors  alone,  if  corrections  by  the  pencil  may  be 
made  alongside  without  confusion,  as  it  is,  in  such  a case,  time 
enough  to  clear  away  superfluous  lines  when  the  inking  is  finished. 

When  ink  lines  to  any  considerable  extent  have  to  be  erased,  a 
small  piece  of  damped  soft  sponge  may  be  rubbed  over  them  till 
they  disappear.  As,  however,  this  process  is  apt  to  discolor  the 
paper,  the  sponge  must  be  passed  through  clean  water,  and  applied 
again  to  take  up  the  straggling  ink.  For  small  erasures  of  ink  lines, 
a sharp  erasing  knife  should  be  used;  this  is  an  instrument  with  a 
short  triangular  blade  fastened  to  a wooden  or  ivory  handle.  A 
sharp  rounded  pen-blade  applied  lightly  and  rapidly  does  well,  and 
the  surface  may  be  smoothed  down  by  the  thumb  nail  or  a paper- 
knife  handle.  In  ordinary  working  drawings  a line  may  readily  be 
taken  out  by  damping  it  with  a hair  pencil  and  quickly  applying  the 
india-rubber;  and,  to  smooth  the  surface  so  roughened,  a light  ap- 
plication of  the  knife  is  expedient.  In  drawings  intended  to  be 
highly  finished,  particular  pains  should  be  taken  to  avoid  the  neces- 
sity for  corrections,  as  everything  of  this  kind  detracts  from  the  ap- 
pearance. 

In  using  the  square  the  more  convenient  way  is  to  draw  lines  ofl 
the  left  edge  with  the  right  hand,  holding  the  stock  steadily,  but 
not  tightly,  against  the  edge  of  the  board  with  the  left  hand.  The 
convenience  of  the  left  edge  for  drawing  by  is  obvious,  as  we  are 
able  to  use  the  arms  more  freely,  and  we  see  exactly  what  we  are 
doing. 

To  draw  lines  in  ink  with  the  least  amount  of  trouble  to  himself, 
the  draftsman  ought  to  take  the  greater  amount  of  trouble  with 
his  tools.  If  they  be  well  made,  and  of  good  stuff  originally,  they 
ouuht  to  last  through  three  generations  of  draftsmen  : their  working 
oarts  should  be  carefully  preserved  from  injury  they  should  be 
kept  well  set,  and  above  all,  scrupulously  clean.  The  setting  of  in- 
struments is  a matter  of  some  nicety,  for  which  purpose  a small 
oil-stone  is  convenient.  To  dress  up  the  tips  of  the  blades  of  the 
pen,  or  of  the  bows,  as  they  are  usually  worn  unequally  by  the  cus* 
tomory  usage,  they  maybe  screwed  up  into  contact  in  the  first. 


134 


PRACTICAL  CARPENTRY. 


place,  and  passed  along  the  stone,  turning  upon  the  point  in  a di 
rectly  perpendicular  plane,  till  they  acquire  an  identical  profile. 
Being  next  unscrewed,  and  examined  to  ascertain  the  parts  of  un- 
equal thickness  round  the  nib,  the  blades  are  laid  separately  upon  their 
back  on  the  stone  and  rubbed  down  at  the  points  till  they  are  brought 
up  to  an  edgs  of  uniform  fineness.  It  is  well  to  screw  them  together 
again,  and  to  pass  them  over  the  stone  once  or  twice  more,  to  bring 
up  any  fault;  to  retouch  them  also  on  the  outer  and  inner  side  of 
each  blade,  to  remove  barbs  or  frazing;  and  finally  to  draw  them 
across  the  palm  of  the  hand. 

The  India-ink,  which  is  commonly  used  for  line-drawing,  ought 
to  be  rubbed  down  in  water  to  a certain  degree — avoiding  the 
sloppy  aspect  of  light  lining  in  drawings,  and  making  the  ink  just 
so  thick  as  to  run  freely  from  the  pen.  This  medium  degree  may  be 
judged  of  after  a little  practice  by  the  appearance  of  the  ink  on 
the  pallet  The  best  quality  of  ink  has  a soft  feel,  free  from  grit 
or  sediment  when  wetted  and  rubbed  against  the  teeth,  and  it  has 
a musky  smell.  The  rubbing  of  India-ink  in  water  tends  to  crack 
and  break  away  the  surface  at  the  point;  this  may  be  prevented  by 
shifting  at  intervals  the  position  of  the  stick  in  the  hand  while  being 
rubbed,  and  thus  rounding  the  surface.  Nor  is  it  advisable,  for  the 
same  reason,  to  bear  very  hard,  as  the  mixture  is  otherwise  more 
evenly  made,  and  the  enamel  of  the  pallet  is  less  rapidly  worn  ^off. 
When  the  ink,  on  being  rubbed  down,  is  likely  to  be  for  some  time 
required,  a considerable  quantity  of  it  should  be  prepared,  as  the 
water  continually  vaporizes;  it  will  thus  continue  for  a longer  time 
in  a condition  fit  for  application.  The  pen  should  be  levelled  in 
the  ink,  to  take  up  a sufficient  charge;  and  to  induce  the  ink  to 
enter  the  pen  freely,  the  blades  should  be  lightly  breathed  upon  or 
wetted  before  immersion.  After  each  application  of  ink  the  outsides 
of  the  blades  should  be  cleaned,  to  prevent  any  deposit  of  ink  upon 
the  edge  of  the  square 

To  keep  the  blades  of  his  bikers  clean  is  the  first  duty  of  a 
draftsman  who  is  to  make  a good  piece  of  work.  Pieces  of  blot- 
ting or  unsized  paper,  and  cotton  velvet,  washleather,  or  even  the 
sleeve  of  a coat,  should  always  be  at  hand  while  a drawing  is  being 


PRACTICAL  CARPENTRY. 


*35 

inked.  When  a small  piece  of  Plotting  paper  is  fo’ded  twice,  so  as 
to  present  a corner,  it  may  usefully  be  passed  between  the  blades 
of  the  pen,  now  and  then,  as  the  ink  is  liable  to  deposit  at  the 
point  and  obstruct  the  passage,  particularly  in  fine-lining ; and  for 
this  purpose  the  pen  must  be  unscrewed  to  admit  the  paper.  But 
this  process  may  be  delayed  by  drawing  the  point  of  the  pen  over 
a piece  of  velvet,  or  even  over  the  surface  of  thick  blotting  paper ; 
either  method  clears  the  point  for  a time.  As  soon  as  any  obstruc- 
tion takes  place  the  pen  should  be  immediately  cleaned,  as  the 
trouble  thus  taken  will  always  improve  and  expedite  the  work.  If 
the  pen  should  be  laid  down  for  a short  time  with  the  ink  in  it,  it 
should  be  unscrewed  to  keep  the  points  apart,  and  so  prevent  de- 
posit; and  when  done  with  altogether  for  the  occasion,  it  ought  to 
be  thoroughly  cleaned  at  the  nibs.  This  will  prevent  rusting,  and 
preserve  its  edges. 

In  coloring  drawings  it  is  necessary  to  have  good  brushes,  or 
camel’s-hair  pencils  for  the  purpose  ; these  can  be  purchased  from 
any  dealer  who  keeps  painter’s  materials.  The  point  of  each  brush 
should  be  as  fine  as  the  point  of  a small  needle,  and  when  the 
brush  is  bent  over  after  being  wetted,  the  hairs  should  remain  in  a 
compact  bunch  without  any  separation  whatever.  If  it  should  split 
up  into  several  parts  during  the  operation  of  bending  to  and  fro,  it 
should  be  examined,  and  all  short  hairs  found  in  the  centre  removed* 
Bright  clear  lines  are  made  by  having  the  ink  very  dark . 

Never  intersect  lines  until  the  first  lines  drawn  are  dry. 

Tapering  lines  are  made  by  successive  closings  of  the  pen,  or  by  a 
dexterity  in  adjusting  its  position  to  the  paper  and  the  pressure  upon  it. 

When  it  is  intended  to  tint  drawings  with  ink  or  colors,  the  fob 
lowing  rules  should  be  observed:  (i)  The  paper  should  have 

the  superfluous  sizing  removed  by  being  sponged  lightly  with  clean 
water.  (2)  The  paper,  and  everything  about  it,  must  be  kept  per- 
fectly clean.  (3)  Line  off  the  spaces  with  very  fine  pencil  marks, 
that  are  to  be  tinted.  (4)  Never  use  the  eraser  on  the  part  to  be 
tinted,  either  before  or  after  the  tinting.  (5)  Try  the  tinting  process 
on  a piece  of  waste  paper  until  the  proper  tint  is  obtained,  before 
applying  to  the  drawing.  (6)  Dark  tints  are  formed  by  applying 


136 


PRACTICAL  CARPENTRY* 


a number  of  light  ones  over  each  other,  but  a second  tint  should 
notvbe  applied  until  the  first  one  is  perfectly  dry.  (7)  Always  finish 
tinting  one  portion  of  drawing  before  leaving  it.  Otherwise  it  will 
be  cloudy.  (8)  See  that  the  paper  is  damp  before  you  begin  to 
tint.  (9)  Ink  in  all  lines  after  the  tinting  is  completed  and  the 
drawing  is  perfectly  dry. 

The  following  arrangement  shows  how  the  various  materials  are 
represented  in  a drawing,  but  in  my  experience  I have  always  found 
it  the  better  way  to  leave  all  working  drawings  in  ink  alone.  An 
ink  drawing  nicely  finished,  has  a far  more  artistic  appearance  than 
one  that  is  colored,  unless  the  latter  is  made  by  a master  hand. 


Materials.  Color . 

Brickwork  in  plans  and  sections. Crimson  lake. 

Brickwork  in  elevations Crimson  lake  mixed  with  burnt 

sienna  or  Venetian  red. 

The  lighter  woods,  such  as  pine. Raw  sienna. 

Oak  or  ash Vandyke  brown. 

Granite Pale  Indian  ink. 

Stone  generally Yellow  ochre,  or  pale  sepia. 

Concrete  works Sepia  with  darker  markings. 

Wrought  iron  . : Indigo. 

Cast  iron Payne’s  grey  or  neutral  tint. 

c;tee] Pale  indigo  tinged  with  lake. 

Brass Gamboge  or  Roman  ochre. 

Lead Pale  Indian  ink  tinged  with  indigo. 

Clay  or  earth Burnt  umber. 

Slate Indigo  and  lake. 


DRAWING  PAPER. 

Drawing  paper,  properly  so  called,  is  made  to  certain  standard 
sizes,  as  follows : 

Demy 20  X 15  inches. 

Medium 22X17 

Royal 24  X 17 


(i 


PRACTICAL  CARPENTRY. 


*37 


Super-  R.oyal 

inches. 

Imperial 

i< 

Elephant 

u 

Columbia 

34  x 23 

u 

Atlas 

34  X 26 

u 

Double  Elephant  .... 

40  X 27 

u 

Antiquarian 

53X31 

u 

Emperor 

. . . . 68  X 48 

u 

Of  these,  Double  Elephant  is  the  most  generally  useful  size  of 
sheet.  Demy  and  Imperial  are  the  other  useful  sizes.  Whatman’s 
white  paper  is  the  quality  usually  employed  for  finished  drawings; 
it  will  bear  wetting  and  stretching  without  injury,  and,  when  so 
treated,  receives  shading  and  coloring  easily  and  freely.  Common 
papers  will  answer  for  drawings  where  no  damping  or  stretching  is 
required.  Large  drawings,  that  are  frequently  referred  to,  should  be 
mounted  on  linen,  previously  damped,  with  a free  application  of 
paste. 

Drawing  paper  also  comes  in  rolls  of  indefinite  lengths,  and  from 
36  to  54  inches  wide.  It  is  made  different  tints,  is  generally  very 
tough,  and  is  chiefly  used  for  details;  it  is  much  cheaper  than 
Whatman’s,  and  for  many  purposes  answers  just  as  well.  Tracing 
cloth,  also,  comes  in  rolls,  18,  30,  36,  and  42  inches  wide;  it  is 
convenient  and  durable,*  and  may  be  folded  up  almost  any  number 
of  times  without  injury. 

Tracing  paper  is  made  of  different  qualities  and  sizes ; it  is 
rendered  transparent,  and  qualified  to  receive  ink  lines  and  tinting 
without  spreading.  Like  tracing  cloth,  when  placed  over  a draw- 
ing already  executed,  the  drawing  is  distinctly  visible  through  the 
paper,  and  may  be  copied  or  traced  directly  by  the  ink  instru- 
ments; thus  an  accurate  copy  may  be  made  with  great  expedition. 

BOARD  MEASURE. 

Explanation  of  Table. — The  left-hand  column  shows  the  width 
of  each  board  in  inches.  At  the  heads  of  the  remaining  columns 
will  be  found  the  lengths  of  the  boards  in  feet.  The  contents  of 
each  board  is  given  in  square  feet  and  primes  or  12th  parts  of  a 


PRACTICAL  CARPENTRY. 


138 

square  foot,  1 prime  being  equal  to  12  square  inches.  For  exam- 
ple, a board  8 inches  wide  and  4 feet  long  contains  2 square  feet 
and  8 primes , or  96  square  inches. 

Operation. 

Bringing  4 feet  to  inches  thus:  4 X 12  =48  inches.  Then 

'48  X 8 = 384.  Lastly  : 144)384(2 

288 

96 

Thus  a board  4 feet  long  and  8 inches  wide  contains  2 square 
feet,  and  8-12  of  a square  foot  or  96  square  inches.  In  construct- 
ing the  following  table  the  remaining  square  inches  (after  dividing 
by  144  to  bring  the  inches  to  feet)  are  divided  by  12,  and  the 
quotient  is  termed  primes. 

Boards  are  sold  by  superficial  measurement,  at  so  much  per  foot 
of  an  inch  or  less  in  thickness ; adding  one-fourth  to  the  price  for 
each  quarter  of  an  inch  thickness  over  an  inch.  It  sometimes 
happens  that  a board  is  tapering,  being  wider  at  one  end  than  the 
other.  When  this  is  the  case  (if  it  be  a true  taper),  add  the  width 
of  both  ends  together,  and  half  their  sum  will  express  the  average 
width  of  the  board.  Again  : if  the  board  does  not  taper  regularly, 
take  the  following  course  to  find  its  area:  I.  Measure  the  breadths 

at  several  places  equi-distant.  II.  Add  together  the  different 
breadths,  and  half  the  two  extremes.  III.  Multiply  this  sum  by 
the  straight  side  of  the  board  and  divide  the  product  by  the  number 
of  parts  into  which  the  board  was  divided.  It  is  usual  in  measur- 
ing rough  lumber,  to  pay  no  attention  to  fractions  of  an  inch  in  the 
width  of  the  stuff.  If  the  fraction  is  more  than  half  an  inch  it  is 
counted  as  an  inch ; if  less  than  half  an  inch  it  is  not  counted. 
Thus,  a board  10^4  inches  wide,  would  be  measured  as  1 1 inches 
wide;  if  only  1 6}i  inches  wide  the  board  will  pass  only  as  10 
inches  wide.  When  the  fraction  is  just  a half  an  inch,  it  is  some- 
times counted  and  sometimes  not.  Local  usage,  however,  fome- 
times  runs  contrary  to  this  practice,  and  the  fractions  are  not  taken 
into  consideration  at  all.  The  widths  are  all  measured  by  whole 
numbers. 


PRACTICAL  CARPENTRY, 


*39 


LENGTH  IN  FEET. 


Inches 

in 

Width. 

3 feet. 

4 feet. 

5 feet. 

6 feet. 

7 feet. 

8 feet. 

q feet*. 

1 

0 

3 

0 4 

0 5 

0 6 

07 

0 8 

09 

2 

0 

6 

08 

0 10 

10 

12 

14 

16 

3 

0 

9 

10 

13 

16 

1*9 

2 0 

2*3 

4 

1 

0 

1 4 

1-8 

2 0 

2*4 

28 

3 0 

5 

1 

3 

1*8 

2 1 

2-6 

211 

3.4 

3 9 

6 

1 

6 

20 

2-6 

30 

3*6 

4 0 

4-6 

7 

1 

9 

2-4 

211 

3*6 

41 

4*8 

5 3 

8 

2 

0 

2*8 

3 4 

40 

4-8 

5-4 

60 

9 

2 

3 

30 

3 9 

4-6 

5 3 

60 

6*9 

10 

2 

6 

34 

4 2 

50 

5*  10 

68 

7 6 

11 

2 

9 

3*8 

4 7 

5-6 

6-5 

7*4 

8*3 

12 

3 

0 

40 

50 

60 

70 

80 

90 

13 

3 

3 

4*4 

5-5 

6*6 

77 

8*8 

9-9 

14 

3 

6 

48 

510 

7 0 

82 

94 

10*6 

15 

3 

9 

5 0 

6-3 

7-6 

89 

100 

11*3 

16 

4 

0 

5*4 

68 

80 

9-4 

10*8 

12*0 

17 

4 

3 

5-8 

71 

8-6 

911 

11*4 

12*9 

18 

4 

6 

60 

•76 

90 

10  6 

120 

13*6 

19 

4 

9 

6-4 

7 11 

96 

11  1 

12-8 

14*3 

20 

5 

0 

6-8 

8-4 

100 

11-8 

13  4 

15  0 

21 

5 

3 

70 

8 9 

10-6 

12*3 

14  0 

15-9 

22 

5 

6 

7*4 

9-2 

110 

1210 

14  8 

16  6 

23 

5 

9 

7 8 

9-7 

11*6 

13*5 

15  4 

17  3 

24 

6 

0 

80 

100 

12  0 

14  0 

160 

18  1 

25 

6 

3 

8*4 

10  5 

12  6 

14-7 

16*8 

18*9 

26 

6 

6 

8*8 

10  10 

13  0 

15  2 

17  4 

19  6 

27 

6 

9 

9 0 

113 

13-6 

159 

180 

20  3 

28 

7 

0 

9*4 

11*8 

140 

16*4 

18*8 

210 

29 

7 

3 

9*8 

12  1 

14-6 

1611 

19*4 

21*9 

30 

7 

6 

10  0 

12  6 

150 

17*6 

200 

22*6 

31 

7 

9 

10-4 

1211 

15-6 

18  1 

20  8 

23  3 

32 

8 

0 

10  8 

13  4 

160 

18-8 

21  4 

24  0 

33 

8 

3 

no 

13-9 

16-6 

19  3 

22  0 

24*9 

34 

8 

6 

11*4 

14  2 

17  0 

1910 

22  8 

25  6 

35 

8 

9 

118 

14  7 

17  6 

20  5 

234 

26*3 

36 

9 

0 

120 

15  0 

18  0 

210 

24  0 

270 

140 


PRACTICAL  CARPENTRY, 


LENGTH  IN  FEET. 


Inches 

in 

Width. 

xofeet. 

1 1 feet. 

12  feet. 

13  feet. 

14  feet. 

15  feet. 

16  feet. 

17  feet. 

1 8 feet 

1 

0*10 

Oil 

10 

11 

1-2 

1*3 

11 

1*5 

1-6 

2 

1*8 

110 

2 0 

2-2 

21 

2*6 

2-8 

210 

30 

3 

2-6 

2-9 

3-0 

3*3 

3-6 

3*9 

4-0 

4-3 

4-6 

4 

3-4 

3*8 

4-0 

4*4 

4-8 

50 

51 

5-8 

6-0 

5 

4-2 

4*7 

5*0 

5*5 

510 

6*3 

6-8 

71 

7*6 

6 

5-0 

5*6 

6*0 

6*6 

7-0 

76 

8-0 

8*6 

9-0 

7 

5*10 

6*5 

7*0 

7-7 

8*2 

8-9 

91 

911 

10-6 

8 

6-8 

7-4 

8-0  , 

8*8 

91 

100 

10-8 

111 

12  0 

9 

7*6 

8-3 

9 0 

9-9 

10-6 

11*3 

120 

12-9 

13-6 

10 

8-4 

9*2 

10-0 

1010 

11-8 

12-6 

131 

14'2 

150 

11 

9-2 

101 

no 

1111 

1210 

13-9 

14*8 

15-7 

16-6 

12 

100 

110 

120 

130 

14-0 

150 

160 

17-0 

18*0 

13 

1010 

1111 

13*0 

141 

15-2 

16-3 

171 

18*5 

19*6 

14 

11*8 

1210 

140 

15-2 

161 

17;  6 

18*8 

1910 

21-0 

15 

12-6 

13-9 

15*0 

16*3 

17*6 

18*9 

20-0 

21-3 

22-6 

16 

13-4 

14-8 

16*0 

171 

18-8 

20*0 

211 

22-8 

24-0 

17 

142 

15*7 

170 

18-5 

1910 

21-3 

22*8 

241 

25*6 

18 

150 

16-6 

18-0 

19-6 

21-0 

22-6 

24-0 

25*6 

27*0 

19 

1510 

l7  5 

19 ’0 

20-7 

22-2 

23-9 

251 

2611 

28-6 

20 

16-8 

18 -4 

20-0 

21-8 

231 

25-0 

26*8 

281 

30-0 

21 

17-6 

19-3 

21-0 

22*9 

24-6 

26-3 

280 

29-9 

31-6 

22 

18-4 

20-2 

22  0 

2310 

25-8 

27*6 

291 

31*2 

33-0 

23 

19-2 

211 

23-0 

2411 

2610 

28*9 

30-8 

32-7 

34-6 

24 

20  0 

22  or 

240 

26  0 

28-0 

30*0 

320 

34*0 

36*0 

25 

2010 

2211 

25-0 

271 

29-2 

31-3 

331 

35-5 

37*6 

26 

21  8 

2310 

260 

28-2 

301 

32*6 

34-8 

3610 

39-0 

27 

22-6 

24-9 

27-0 

29*3 

316 

33-9 

360 

38*3 

40-6 

28 

23*4 

25*8 

28-0 

301 

32-8 

35  0 

371 

39*8 

42-0 

29 

24 ‘2 

26-7 

29-0 

31*5 

3310 

36-3 

38-8 

411 

43-6 

30 

25  0 

27*6 

30  0 

32-6 

35  0 

37*6 

40*0 

42-6 

45-0 

31 

2510 

28-5 

310 

33-7 

36-2 

38*9 

411 

4311 

46*6 

32 

26-8 

29-4 

32-0 

34*8 

371 

40-0 

42 -8 

451 

48*0 

33 

27*6 

30-3 

33-0 

35*9 

38-6 

41*3 

44-0 

46*9 

49*6 

34 

28-4 

31-2 

34-0 

3610 

39-8 

42-6 

451 

48-2 

51*0 

35 

29 '2 

321 

35-0 

3711 

4010 

43*9 

46-8 

49-7 

52*6 

36 

30  0 

33-0 

36-0 

39*0 

42-0 

45  0 

48*0 

51*0 

54;6 

PRACTICAL  CARPENTRY. 


141 


AMERICAN  WEIGHTS  AND  MEASURES. 


Lineal  Measure. 


Cubic  Measure . 


12  inches 1 foot. 

3 feet 1 yard. 

16  ft.  6 ins.  or  5J  y’ds.  1 rod. 

220  yards  or  40  rods. . . 1 furlong 
1760  yards  or  8 furlongs.  1 mile. 

7  22  inches 1 link. 

100  links  or  66  feet 1 chain. 

10  chains 1 furlong. 

80  chains 1 mile. 

69^  statute  miles  or^.  j 1 degree  oi 
60*geographical  miles  j the  equatoi 

8  furlongs 1 mile. 

3 miles 1 league. 


Square  Measure. 

144  square  inches 1 foot. 

9  square  feet 1 yard. 

100  44  44  1 square. 

2™t  “ “ °/  l 1 rod,  pole 

3(H  square  yards  V 

40  square  rods. . . . ) ^ 

4 square  rods  or  j t 

4840  yards...  \ 1 acre‘ 

10,000  square  links 1 sq.  chain 

10  sq.  chains  100,000  I , 

sq.  links \ ls<l-acre- 

640  acres  1 sq.  mile. 


1728  cubic  inches 

27  cubic  feet 

40  feet  of  round  or  | 
50  ft.  of  hewn  timb.  J 
45  cubic  feet 

8 cord  feet  or  ) 

128  cubic  feet. . j 


1 cubic  ft. 
1 cubic  yd. 

1 ton. 

1 ton  of 
shipping. 

1 cord. 


Circular  Measure. 

60  seconds 1 minute. 

60  minutes I degree. 

15  degrees 1 hour  angle 

30  degrees 1 sign. 

12  signs  or  360  degrees.  1 circle. 

Books. 

A sheet  folded  in  2 leaves  a folio. 


«< 

4 •*  j 

i a quarto 
i or  4to. 

• « 

8 - j 

1 an  octavo 
j or  8vo. 

II 

12  44 

a 12  mo. 

II 

16  44 

a 16  mo. 

II 

18  44 

a 18  mo. 

II 

24  44 

a 24  mo. 

II 

32  44 

a 32  mo. 

DECIMAL  APPROXIMATIONS  FOR  FACILITATING  CALCULATIONS. 


Lineal  feet  multiplied  by 
“ yard3  44 

Square  inches  44 

44  feet  44 

44  yards  44 

Circular  inches  44 

44  feet 
Cubic  inches  44 

44  feet 

«i  «*  I 4 

44  inches  44 

Bushels  •• 

««  t» 


•00019  .=  miles. 

•000598  = 44 

•007  = square  feet. 

•Ill  = square  yards, 
•0002067  = acres. 

•00546  = square  feet. 

02909  = cubic  yards. 

•00058  = cubic  feet. 

•03704  — cubic  yards. 

6*2321  ==  imperial  gallons 

•003607  = 44  44 

0476  — cubic  yards, 

1-284  ==  cubic  feet. 


142 


PRACTICAL  CARPENTRY. 


AUTHORIZED  METRIC  SYSTEM. 

Measures  of  Length. 


Metric  Denominations  and  Values. 


10,000 

metres 

1,000 

metres 

100 

metres 

10 

metres 

1 

metre 

1-10 

of  a metre 

1-100 

of  a metre 

1-1000 

of  a metre 

Equivalents  in  Denominations  in  Use. 


Myriametre . 
Kilometre. . . 

Hectometre.. 
Decametre . 

Metre 

Decimetre  . 
Centimetre. . 
Millimetre. . 


6*2137  miles. 

0-G2137  miles,  or  3,280  feet 
10  inches. 

328  feet  and  1 inch. 

393  7 inches. 

39-37  inches. 

3*937  inches. 

0 3937  inch. 

0 0394  inch. 


Measures  of  Surfaces. 


Metric  Denomm  -tio  >s  and  Values 

Equivalents  in  Denominations  in  Use 

Hectare 

10,000  square  metres 

2 *47 1 acres. 

Arc 

100  square  metres 

119*6  square  yards. 

Centare 

1 square  metre  . . . 

1550  square  inches. 

Measures  of  Capacity. 


Metric  Denominations  and  Values. 


Equivalents  in  Denominations  in  Use. 


Names. 

No.  of" 
litres 

Cubic  Measure. 

Dry  Measure. 

Liquid  or  Wine 
Measure. 

Kilolitre,  or  stere. . 

Hectolitre 

Decalitre 

Litrp:  

Decilifre 

Centilitre 

Millilitre 

1,000 

IOO 

lO 

i 

I-TO 

I-IOO 

I-IOOO 

i cubic  metre 

i-io  of  a cubic  metre 

io  cubic  decimetres 

t cubic  decimetre 

i-io  of  a cubic  decimetre 

io  cubic  centimetres 

i cubic  centimetre  

i -308 cubic  yard. . 
2 bush.  3-35  pk. . . 

9-08  quarts 

0 908  quart 

6 1022  cub.  in  . . 
0 6T02  cub.  in. . . . 
0*061  cub.  in 

264*17  gallons. 
26*417  gallons. 

2 6417  gallons 
1-0567  quarts. 

0 845  gill. 

0 338  fluid  0 z. 

0 27  fluid  dr. 

PRACTICAL  CARPENTRY, 


143 


Weights. 


Metric  Denominations  and  Values. 


Equivalents  in 
Denominations  in  LKe. 


Names. 


Millier,  or  tonneau. . . 

Quintal 

Myriagramme 

Kilogramme,  or  kilo. . . 

Hectogramme 

Decagramme 

Gramme 

Decigramme 

Centigramme 

Milligramme 


No  of 
grammes. 

Weight  of  what  quantity  of 
water  at  maximum  density. 

1,000,000 

1 cubic  metre 

100,000 

1 hectolitre 

I O OOO 

10  litres 

1,000 

1 litre 

100 

1 decilitre 

10 

xo  cubic  centimetres 

1 

1 cubic  centimetres 

I-IO 

i-xo  of  a cubic  centimetre 

I-IOO 

10  cubic  millimetres 

I -IOOO 

1 cubic  millimetre 

Avoirdupois 

Weight. 


2204*6  pounds. 
220*46  pounds 
22  046  pounds. 
2*2046  pounds. 
3*5274  ounces. 
0*3527  ounce... 
15*432  grains. . 
1 *5432  grain... 
o*i 543  grain... 
0*0154  grain  . . 


GEOGRAPHICAL  OR  NAUTICAL  MEASURE. 

6 feet  = 1 fathom. 

110  fathoms,  or  660  feet  = 1 furlong. 

6075  1-5  feet  = 1 nautical  mile. 

3 nautical  miles  = 1 league. 

20  leagues,  or  60  geographical  miles  = 1 degree. 

ocf\  ( the  circumference  of  the  earth, 

360  JeSreea  = \ or  24,855*  miles,  nearly. 

The  nautical  mile  exceeds  the  common  one  by  795  4-5  feet. 


MEASURE  OF  TIME. 


60  seconds  = 1 minute. 
60  minutes  = 1 hour. 

24  hours  = 1 day. 

7 days 
28  days 


28,  29,  30,  or  31  days  = 1 cal.  month. 
12  calendar  months  = 1 year. 

365  days  = 1 com.  year. 

1 week.  366  days  = 1 leap  year. 

1 lunar  month.  365^  days  = 1 Julian  year. 

365  days,  5 hours,  48  minutes,  49  seconds  = 1 solar  year. 

365  days,  G hours,  9 minutes,  12  seconds  = 1 siderialyear. 


ON  CONTRACTS  AND  SPECIFICATIONS. 

It  frequently  happens,  in  villages  and  country  towns,  that  the 
carpenter  will  not  only  be  called  upon  to  make  drawings  or  pencil 
sketches  of  cottages,  barns,  stables,  fences,  etc.,  etc.,  but  he  will  be 
expected  to  make  specifications  for  same  with  bills  of  lumber  and 
all  the  necessary  details  connected  therewith.  A form  for  a speci- 
fication would  take  up  too  much  space  here  to  be  given,  and  it  is 
quite  unnecessary,  for  printed  forms  can  be  purchased  for  35  cents 
each,  which  can  be  filled  in  in  a very  short  time.  It  is  much  better 
for  the  country  workman  to  use  these  forms  in  every  case,  as  they 


144 


PRACTICAL  CARPENTRY* 


will  prove  reminders  of  many  things  that  might  otherwise  be 
forgotten. 

Though  a form  of  specification  may  not  be  necessary  for  the 
reason  given,  yet  it  is  thought  that  a short  pithy  form  of  contract 
might  prove  useful,  as  frequently  there  is  no  other  agreement  in 
writing  between  workmen  and  owners  than  a short  written  con* 
tract.  The  following  may  be  changed  to  suit  conditions : 

POEM  OF  CONTRACT  FOR  BUILDING. 

Made  the  day  of  — — , one  thousand  eight  hundred  and 

— — , by  and  between  — — — , of  the  second  part,  in 

these  words;  the  said  — part of  the  second  part  covenant, 

and  agree  to, and  with  the  said  party  of  the  first  part,  to  make,  erect, 
build,  and  finish,  in  a good  substantial,  and  workmanlike  manner, 

on  the  — — - — — — — — — — agreeable 

to  the  draft,  plan,  and  explanation  hereto  annexed,  of  good  and  sub- 
stantial material,  by  the day  of  — next.  And 

the  party  of  the  first  part  covenants  and  agrees  to  pay  unto  the  said 

« part — of  the  second  part,  for  the  same,  the  sum  cf 

— — — — — — — -,  lawful  money  of  the  United 

States,  as  follows  : The  sum  of : , and  for  the 

true  and  faithful  performance  of  ail  and  every  of  the  covenants  and 
agreements  above  mentioned,  the  parties  to  these  presents  bind  them- 
selves each  unto  the  other,  in  the  penal  sum  of  — — 

dollars;  as  affixed  and  settled  damages  to  be  paid  by  the  failing  party. 

In  witness  whereof,  the  parties  to  these  presents  have  hereunto  set 
their  hands  and  seals,  the  day  and  year  above  written. 


Sealed  and  delivered  in  the  presence  of 


&9115T8  ARB' STTM?®* 


JOINTS  AND  STRAPS. 


PLATE  1L. 


JOINTS* 

fCAftrm*  L£MGTHCNme-ftfAH*jlNP 


PLATE  nt. 


. 


A <— *•  t =£. 


£===!===£ 


SLATE  VT. 


JfyA-MZ, 


JTf.S.MA 


JfyJJUb 


Zu,.6LMJL 


SUPPLEMENT. 


THE  STEEL  SQUARE. 


SOME  DIFFICULT  PROBLEMS 


In  Carpentry  and  Joinery 

SIMPLIFIED  AND  SOLVED  BY  THE  AID  OF  THE 

Carpenters’  Steel  Square 

TOGETHER  WITH  A FULL  DESCRIPTION  OF  THE  TOOL,  AND 
EXPLANATIONS  OF  THE  SCALES,  LINES  AND  FIGURES 
ON  THE  BLADE  AND  TONGUE,  AND  HOW  TO 
USE  THEM  IN  EVERYDAY  WORK. 


Showing  how  the  Square  may  be  Used  in  Obtaining  the  Lengths 
and  Bevels  of  Rafters,  Hips,  Groins,  Braes,  Brackets, 
Purlins,  Collar-Beams  and  Jack-Rafters.  Also,  Its 
Application  in  Obtaining  the  Bevels  and  Cuts 
for  Hoppers,  Spring  Mouldings,  Octagons, 

Diminished  Styles,  Etc.,  Etc.,  Etc. 


ILLUSTRATED  BY  NUMEROUS  WOOD  CUTS. 


BY 


FRED  T.  HODGSON 


MONTGOMERY  WARD  & COMPANY 

CHICAGO 


CONTENTS 


PAGE 

Preface,  5 

Description  of  the  Square,  - 9 

Board  Measure, 11 

Brace  Rule, ,-13 

Octagonal  Scales, 14 

The  Fence, 15 

Application  of  Square, 17 

To  lay  out  Rafters, 20 

To  Cut  Collar  Beams,  Gables,  Pitches,  etc.,  - - - - 25 

Hip  Rafters,  Cripples,  etc.,  = --  --  --25 

Backing  for  Hips, 30 

How  to  Cut  a Mitre  Box, 31 

Stairs  and  Strings, - >32 

Miscellaneous  Rules, 34 

Board  Measure,  - -35 

Proportion  of  Circles, 37 

Centering  Circles, -39 

How  to  Describe  an  Ellipse,  - - - - - 40 

How  to  Describe  a Parabola,  - - - ' - - - -41 

Bevels  for  Hoppers, 42 

Bisecting  Circles, -43 

Octagons,  - ,-  --  --  --  -45 

To  Cut  Raking  Cornice,  -----.--46 
To  Cut  Hopper  Bevels, - 


DESCRIPTION 


OF  THE  SQUARE. 


The  lines  and  figures  formed  on  the  squares  of  different 
make,  sometimes  vary,  both  as  to  their  position  on  the 
square,  and  their  mode  of  application,  but  a thorough 
understanding  of  the  application  of  the  scales  and  lines 
shown  on  any  first-class  tool,  will  enable  the  student  to 
comprehend  the  use  of  the  lines  and  figures  exhibited  on 
other  first-class  squares. 

To  insure  good  results,  it  is  necessary  to  be  careful  m 
the  selection  of  the  tool.  The  blade  of  the  square  should 
be  24  inches  long,  and  two  inches  wide,  and  the  tongue 
from  14  to  18  inches  long 'and  1 y2  inches  wide.  The 
tongue  should  be  exactly  at  right  angles  with  the  blade, 
or  in  "Other  words  the  “ square”  should  be"  perfectly 
square. 

To  test  this  question,  get  a board,  about  12  or  14  inches 
wide,  and  four  feet  long,  dress  it  on  one  side,  and  true  up  one 
edge  as  near  straight  as  it  is  possible  to  make  it.  Lay  the 
board  on  the  bench,  with  the  dressed  side  up,  and  the 
trued  edge  towards  you,  then  apply  the  square,  with  the 


10 


THE  STEEL  SQUARE 


blade  to  the  left,  and  mark  across  the  prepared  board  with 
a penknife  blade,  pressing  close  against  the  edge  of.  the 
tongue ; this  process  done  to  your  satisfaction,  reverse  the 
square,  and  move  it  until  the  tongue  is  close  up  to  the 
knife  mark ; if  you  find  that  the  edge  of  the  tongue  and 
mark  coincide,  it  is  proof  that  the  tool  is  correct  enough 
for  your  purposes. 

This,  of  course,  relates  to  the  inside  edge  of  the  blade, 
and  the  outside  edge  of  the  tongue  If  these  edges  should* 
not  be  straight,  or  should  not  prove  perfectly  true,  they 
should  be  filed  or  ground  until  they  are  straight  and  true. 
The  outside  edge  of  the  blade  should  also  be  “ trued  ” up 
and  made  exactly  parallel  with  the  inside  edge,  if  such 
is  required.  The  same  process  should  be  gone  through 
on  the  tongue.  As  a rule,  squares  made  by  firms  of 
repute  are  perfect,  and  require  no  adjusting;  neverthe- 
less, it  is  well  to  make  a critical  examination  before  pur- 
chasing. 

The  next  thing  to  be  considered  is  the  use  of  the  figures, 
lines,  and  scales,  as  exhibited  on  the  square.  It  is  sup- 
posed that  the  ordinary  divisions  and  sub-divisions  of  the 
inch,  into  halves,  quarters,  eighths,  and  sixteenths  are  un- 
derstood by  the  student;  and  that  he  also  understands 
how  to  use  that  part  of  the  square  that  is  sub-divided  into, 
twelfths  of  an  inch.  This  being  conceded,  we  now  pro- 
ceed to  describe  the  various  rules  as  shown  on  all  good 
squares. 

Description. — On  the  Frontispiece  we  show  both  sides 
of  one  of  the  best  squares  in  the  market.  It  is  known  to 
thetra.de  as  “ No.  ioo,”  and  this  number  will  be  found 


and  Its  uses. 


U 


stamped  always  on  the  face  side  of  the  square  at  the  junc- 
tion of  the  tongue  and  blade.  The  following  instructions 
refer  to  the  Frontispiece  and  accompanying  cuts. 

The  diagonal  scale  is  on  the  tongue  at  the  junction  with 
blade,  Fig.  i,  and  is  for  taking  off  hundredths  of  an  inch. 
The  lengths  of  the  lines  between  the  diagonal  and  the 
perpendicular  are  marked  on  the  latter.  Primary  divisions 
are  tenths,  and  the  junction  of  the  diagonal  lines  with  the 
longitudinal  parallel  lines  enables  the  operator  to  obtain 
divisions  of  one  hundredth  part  of  an  inch ; as,  for  example, 
if  we  wish  to  obtain  twenty-four  hundredths  of  an  inch,  we 
place  the  compasses  on  the  " dots  ” on  the  fourth  parallel 
line,  which  covers  two  primary  divisions,  and  a fraction,  or 
four  tenths,  of  the  third  primary  division,  which  added 
together  makes  twenty-four  hundredths  of  an  inch.  Again, 
if  we  wish  to  obtain  five  tenths  and  seven  hundredths,  we 
operate  on  the  seventh  line,  taking  five  primaries  and  the 
fraction  of  the  sixth  where  the  diagonal  intersects  the 
parallel  line,  as  shown  by  the  “ dots,”  on  the  compasses, 
and  this  gives  us  the  distance  required. 

The  use  of  this  scale  is  obvious,  and  needs  no  further 
explanation. 

Board  Measure. — Under  the  figure  12,  Fig.  2,  on  the 
outer  edge  of  the  blade,  the  length  of  the  boards,  plank,  or 
scantling  to  be  measured  is  given,  and  the  answer  in  feet 
and  inches  is  found  under  the  inches  in  width  that  the 
board,  etc.,  measures.  For  example,  take  a board  nine 
feet  long  and  five  inches  wide;  then  under  the  figure  12, 
on  the  second  line  will  be  found  the  figure  9,  which  is  the 
length  of  the  board ; then  run  along  this  line  to  the  figure 


I 2 


TEEL  SQUARE 


directly  under  the  five  inches  (the  width  of  the  board),  and 
we  find  three  feet  nine  inches,  which  is  the  correct  answer 
in  “ board  measure,”  If  the  stuff  is  two  inches  thick,  the 
sum  is  doubled ; if  three  inches  thick,  it  is  trebled,  etc.,  etc. 
If  the  stuff  is  longer  than  any  figures  shown  on  the  square, 
it  can  be  measured  by  dividing  and  doubling  the  result. 
This  rule  is  calculated,  as  its  name  indicates,  for  board 
measure,  or  for  surfaces  i inch  in  thickness.  It  may  be 
advantageously  used,  however,  upon  timber  by  multiplying 
the  result  of  the  face  measure  of  one  side  of  a piece  by  its 
depth  in  inches.  To  illustrate,  suppose  it  be  required  to 
measure  a piece  of  timber  25  feet  long,  10x14  inches  in 
size.  For  the  length  we  will  take  12  and  13  feet.  For 
the  width  we  will  take  10  inches,  and  multiply  the  result 
by  14.  By  the  rule  a board  12  feet  long  and  10  inches 


Fig.  3. 


wide  contains  10  feet,  and  one  13  feet  long  and  10  inches 
wide,  10  feet  10  inches.  Therefore,  a board  25  feet  long 
and  10  inches  wide  must  contain  20  feet  and  10  inches. 
In  the  timber  above  described,  however,  we  have  what  is 
equivalent  to  14  such  boards,  and  therefore  we  multiply 
this  result  by  14,  which  gives  2Qi  feet  and  8 inches,  the 
board  measure. 

The  “ board  measure,”  as  tehown  on  the  portion  of  the 


AND  ITS  USES. 


13 


square,  Fig.  3,  gives  the  feet  contained  in  each  board  ac- 
cording to  its  length  and  width.  This  style  of  figuring 
squares,  for  board  measure,  is  going  out  of  date,  as  it  gives 
the  answer  only  in  feet. 


Fig.  3 a. 


Fig.  3 a shows  the  method  now  in  use  for  board  measure. 
This  shows  the  correct  contents  in  feet  and  inches.  It  is 
a portion  of  the  blade  of  the  square,  as  shown  at  Fig.  2,  on 
the  Frontispiece. 

Brace  Rule. “—The  “brace  rule”  is  always  placed  on  the 
tongue  of  the  square,  as  shown  in  the  central  space  at  x, 
Fig.  1. 

This  rule  is  easily  understood;  the  figures  on  the  left  of 
the  line  represent  the  “ run  ” or  the  length  of  two  sides  of  a 
right  angle,  while  the  figures  on  the  right  represent  the 
exact  length  of  the  third  side  of  a right-angled  triangle,  in 
inches,  tenths,  and  hundredths.  Or,  to  explain  it  in  another 
way,  the  equal  numbers  placed  one  above  the  other,  may 
be  considered  as  representing  the  sides  of  a square,  and 


14 


THE  STEEL  SQUARE 


the  third* number  to  the  right  the  length  of  the  diagonal  of 
that  square.  Thus  the  exact  length  of  a brace  from  point 
to  point  having  a run  of  33  inches  on  a post,  and  a run  of 
the  same  on  a girt,  is  46*67  inches.  The  brace  rule  varies 
somewhat  in  the  matter  of  the  runs  expressed  in  different 
squares.  Some  squares  give. a few  brace  lengths  of  which 
the  runs  upon  the  post  and  beam  are  unequal. 

Octagonal  Scale.— The  “ octagonal  scale,”  as  shown  on 
the  central  division  of  the  upper  portion  of  blade,  is  on  the 
opposite  side  of  the  square  to  the  “ brace  rule,”  and  runs 
along  the  centre  of  the  tongue  as  at  s s.  Its  use  is  as  fol- 
lows : Suppose  a stick  of  timber  ten  inches  square.  Make 
a centre  line,  which  will  be  five  inches  from  each  edge ) set 
a pair  of  compasses,  putting  one  leg  on  any  of  the  main 
divisions  shown  on  the  square  in  this  scale,  and  the  other 
leg  on  the  tenth  subdivision.  This  division,  pricked  off 
from  the  centre  line  on  the  timber  on  each  side,  will  give 
the  points  for  the  gauge-lines.  Gauge  from  the  corners 
both  ways,  and  the  lines  for  making  the  timber  octagonal 
in  its  sectiort  are  obtained.  Always  take  the  same  number 
of  spaces  on  your  compasses  as  the  timber  is  inches  square 
from  the  centre  line.  Thus,  if  a stick  is  twelve  inches 
square,  take  twelve  spaces  on  the  compasses ; if  only  six 
inches  square,  take  six  spaces  on  the  compasses,  etc.,  etc. 
The  rule  always  to  be  observed  is  as  follows:  Set  off  from 
each  side  of  the  centre  line  upon  each  face  as  many  spaces 
by  the  octagon  scale  as  the  timber  is  inches  square.  For 
timbers  larger  in  size  than  the  number  of  divisions  in  the 
scale,  the  measurements  by  it  may  be  doubled  or  trebled, 
as  the  case  may  be* 


AND  ITS  USES. 


*5 


I have  now  fully  described  all  the  lines,  figures,  and 
scales  that  are  usually,  found  on  the  better  class  of  squares 
now  in  use ; but,  I may  as  well  here  remark  that  there  are 
squares  in  use  of  an  inferior  grade,  that  are  somewhat  dif- 
ferently figured.  These  tools,  however,  are  such  as  can 
not  be  recommended  for  the  purposes  of  the  scientific 
carpenter  or  joiner. 

Fence. — A necessary  appendage  to  the  steel  square  in 
solving  mechanical  problems,  is,  what  I call,  for  the  want 
of  a better  name,  ah  adjustable  fence.  This  is  made  out 
of  a piece  of  black  walnut  or  cherry  2 inches  wide,  and  2 
feet  10  inches  long  (being  cut  so  that  it  will  pack  in  a tool 
chest),  and  i^s  inches  thick;  run  a gauge  line  down  the 
centre  of  both  edges ; this  done,  run  a saw  kerf  cutting 
down  these  gauge  lines  at  least  one  foot  from  each  end, 
leaving  about  ten  inches  of  solid  wood  in  the  centre  of 
fence.  We  now  take  our  square  and  insert  the  blade  in  the 
saw  kerf  at  one  end  of  the  fence,  and  the  tongue  in  the 
kerf,  at  the  other,  the  fence  forming  the  third  side  of  a 
right-angle  triangle,  the  blade  and  the  tongue  of  the  square 
forming  the  other  two  sides.  A fence  may  be  made  to  do 


pretty  fair  service,  if  the  saw  kerf  is  all  cut  from  one  ena, 
as  shown  at  Fig.  4.  The  one  first  described,  however,  will 
be  found  the  most  serviceable.  The  next  step  will  be  to 


Fig.  4. 


i6 


THE  STEEL  SQUARE 


make  some  provision  for  holding  the  fence  tight  on  tht 
square;  this  is  best  done  by  putting  a No.  io  i*4  inch 
screw  in  each  end  of  the  fence,  close  up  to  the  blade  and 
tongue;  having  done  this,  we  are  ready  to  proceed  to 
business. 

Although  I recommend  a fence,  it  must  be  understood 
that  a fence  is  not  absolutely  necessary,  as  the  work  in 
nearly  all  cases  can  be  performed  without  .it;  though  it 
will  take  more  time  and  care  in  handling  the  square  with- 
out than  with  it.  In  laying  out  rafters  and  braces,  as  will 
be  shown  in  the  following  pages,  it  will  be  better  in  many 
cases  to  apply  the  square  without  using  the  fence,  as  a 
single  brace  or  rafter  may  often  be  “ laid  out  ” and  finished 
in  less  time  than  it  will  take  to  adjust  the  fence.  It  must 
be  borne  in  mind,  however,  that  the  operator  can  make  no 
mistake  when  using  the  square  with  fence  attached,  when 
once  the  arrangement  is  properly  adjusted ; and  it  is  this 
certainty  of  correctness  that  induces  me  to  urge  the  con- 
tinual use  of  the  fence,  wherever  its  application  seems 
necessary. 

I have  known  cases  where  the  fence  and  square  com- 
bined have  been  used  in  lieu  of  a large  bevel,  where  the 
latter  tool  has  not  been  at  hand.  The  intelligent  workman 
will  readily  see  how  this  may  be  accomplished,  as  it  will 
only  be  necessary  for  him  to  adjust  the  square  so  that  one 
of  the  edges  of  the  blade  or  tongue  will  coincide  with  the 
angle  required.  Indeed,  the  arrangement  of  square  and 
fence  may  be  made  to  serve  a thousand  and  one  useful 
purposes,  if  .the  user  possesses  intelligence  enough  to 
master  the  simplest  operations  that  can  be  performed  on 
the  combined  tools. 


AND  ITS  USES. 


17 


Application.— The  fence  being  made  as  desired,  in  either 
of  the  methods  mentioned,  and  adjusted  to  the  square,  work 
can  be  commenced  forthwith. 

The  first  attempt  will  be  to  make  a pattern  for  a brace, 
for  a four-foot  “ run.”  Take  a piece  of  stuff  already  pre- 
pared, six  feet  long,  four  inches  wide  and  half-inch  thick, 
gauge  it  three-eighths  from  jointed  edge. 

Take  the  square  as  arranged  at  Fig.  5,  and  place  it  on 
the  prepared  stuff  as  shown  at  Fig.  6.  Adjust  the  square 
so  that  the  twelve-inch  lines  coincide  exactly  with  the 
gauge  line  o,  o,  o,  o.  Hold  the  square  firmly  in  the  posi- 
tion now  obtained,  and  slide  the  fence  up  the  tongue  and 
blade  until  it  fits  snugly  against  the  jointed  edge  of  the 
prepared  stuff,  screw  the  fence  tight  on  the  square,  and  be 
sure  that  the  1 2 inch  marks  on  both  the  blade  and  the 
tongue  are  in  exact  position  over  the  gauge-line. 

I repeat  this  caution,  because  the  successful  completion 
of  the  work  depends  on  exactness  at  this  stage. 

We  are  now  ready  to  lay  out  the  pattern.  Slide  the 
square  to  the  extreme  left,  as  shown  on  the  dotted  lines  at 
x,  mark  with  a knife  on  the  outside  edges  of  the  square, 
cutting  the  gauge-line.  Slide  the  square  to  the  right  until 
the  12  inch  mark  on  the  tongue  stands  over  the  knife  mark 
on  the  gauge-line  ; mark  the  right-hand  side  of  the  square 
cutting  the  gauge-line  as  before,  repeat  the  process  four 
times,  marking  the  extreme  ends  to  cut  off,  and  we  have 
the  length  of  the  brace  and  the  bevels. 

Square  over,  with  a try  square,  at  each  end  from  the 
gauge-line,  and  we  have  the  toe  of  the  brace.  The  lines, 
s,  s,  shown  at  the  ends  of  the  pattern,  represent  the  tenons 
that  are  to  be  left  on  the  braces.  This  pattern  is  now  com- 


i8 


THE  STEEL  SQUARE 


//  //\\ 


F/G.5 


IF 

"^pQr  — tf! 

m 

CNJ  , r/v^f 

4=j  -jrnZr'  i 
L- 

I wn 

'f':‘Fia.l 

m 

8 

AND  ITS  USES. 


J9 


plete ; to  make  it  handy  for  use,  however,  nail  a strip  2 inches 
wide  on  its  edge,  to  answer  for  a fence  as  shown  at  k,  and 
the  pattern  can  then  be  used  either  side  up. 

The  cut  at  Fig.  7,  shows  the  brace  in  position,  on  a re- 
duced scale.  The  principle  on  which  the  square  works  in 
the  formation  of  a brace  can  easily  be  understood  from  this 
cut,  as  the  dotted  lines  show  the  position  the  square  was  in 
when  the  pattern  was  laid  out. 

It  may  be  necessary  to  state  that  the  “ square,”  as  now 
arranged,  will  lay  out  a brace  pattern  for  any  length,  if  the 
angle  is  right,  and  the  ru?i  equal.  Should  the  brace  be  of 
great  length,  however,  additional  care  must  be  taken  in  the 
adjustment  of  the  square,  for  should  there  be  any  departure 
from  truth,  that  departure  will  be  repeated  every  time  the 
square  is  moved,  and  where  it  would  not  affect  a short  run,  it 
might  seriously  affect  a long  one. 

To  lay  out  a pattern  for  a brace  where  the  run  on  the 
beam  is  three  feet,  and  the  run  down  the  post  four,  proceed 
as  follows: 

Prepare  a piece  of  stuff,  same  as  the  one  operated  on  for 
four  feet  run;  joint  and  gauge  it.  Lay  the  square  on  the 
left-hand  side,  keep  the  12  inch  mark  on  the  tongue,  over 
the  gauge-line,  place  the  9 inch  mark  on  the  blade,  on  the 
gauge-line,  so  that  the  gauge-line  forms  the  third  side  of  a 
right-angle  triangle,  the  other  sides  of  which  are  nine  and 
twelve  inches  respectively. 

Now  proceed  as  on  the  former  occasion,  and  as  shown 
at  Fig.  8,  taking  care  to  mark  the  bevels  at  the  extreme 
ends.  The  dotted  lines  show  the  positions  of  the  square, 
as  the  pattern  is  being  laid  out. 

Fig.  9 shows  the  brace  in  position,  the  dotted  lines  show. 


20 


THE  STEEL  SQUARE 


where  the  square  was  placed  on  the  pattern.  It  is  well  to 
thoroughly  understand  the  method  of  obtaining  the  lengths 
and  bevels  of  irregular  braces.  A little  study,  will  soon 
enable  any  person  to  make  all  kinds  of  braces. 

If  we  want  a brace  with 
a two  feet  run,  and  a four 
feet  run,  it  must  be  evident 
that,  as  two  is  the  half  of 
four,  so  on  the  square  take 
12  inches  on  the  tongue,  and 
6 inches  on  the  blade,  apply 
four  times,  and  we  have  the 
length,  and  the  bevels  of  a 
brace  for  this  run. 

For  a three  by  four  feet 
run,  take  12  inches  on  the 
tongue,  and  9 inches  on  the 
blade,  and  apply  four  times, 
because,  as  3 feet  is  of  four 
feet,  so  9 inches  is  of  12 
inches. 

Rafters. — Fig.  10  shows 
a plan  of  a roof,  having 
twenty-six  feet  of  a span. 

The  span  of  a roof  is  the 
distance  over  the  wall  plates 
measuring  from  A to  A,  as  shown  in  Fig.  10.  It  is  also 
the  extent  of  an  arch  between  its  abutments. 

There  are  two  rafters  shown  in  position  on  Fig.  10.  The 
one  on  the  left  is  at  an  inclination  of"  quarter  pitch,  and 


AND  ITS  USES. 


2t 


marked  B,  and  the  one  on  the  right, 
marked  C,  has  an  inclination  of  one-third 
pitch.  These  angles,  or  inclinations  rather, 
are  called  quarter  and  third  pitch,  respec- 
tively, because  the  height  from  level  of  wall 
plates  to  ridge  of  roof  is  one-quarter  or  one- 
third  the  width  of  building,  as  the  case 
may  be. 

At  Fig.  ii,  the  rafter  B is  shown  drawn  to 
a larger  scale ; you  will  notice  that  this  rafter 
is  for  quarter  pitch,  and  for  convenience,  it  is 
supposed  to  consist  of  a piece  of  stuff  2 
inches  by  6 inches  by  17  feet.  That  portion 
of  the  rafter  that  projects  over  the  wall  of  the 
building,  and  forms  the  eve,  is  three  or  more 
inches  in  width,  just  as  we  please.  The 
length  of  the  projecting  pie^e  in  this  case  is 
one  foot— it  may  be  more  or  less  to  suit  the 
eve,  but  the  line  must  continue  from  end  to 
end  of  the  rafter,  as  shown  on  the  plan,  and 
we  will  call  this  line  our  working  line. 

We  are  now  ready  to  lay  out  this 
rafter,  and  will  proceed  as  follows : We 
adjust  the  fence  on  the  square  the  same  as 
for  braces,  press  the  fence  firmly  against 
the  top  edge  of  rafter,  and  place  the  figure 
i2  inches  on  the  left-hand  side,  and  the 
figure  6 in  on  the  right-hand  side,  directly 
over  the  working  line,  as  shown  on  the 
plan.  Be  very  exact  about  getting  the 
figures  on  the  line,  for  the  quality  oi  the 


THE  STEEL  SQUARE 


22 

woik  depends  much  on  this;  when  you  are  satisfied  that 
you  are  right,  screw  your  fence  tight  to  the  square.  Com- 
mence at  No.  i on  the  left,  and  mark  off  on  the  working 
line;  then  slide  your  square  to  No.  2,  repeat  the  marking 
and  cont:nue  the  process  until  you  have  measured  off 
thirteen  spaces,  the  same  as  shown  by  the  dotted  lines  in 
the  drawing.  The  last  line  on  the  right-hand  side  will  be 
the  plumb  cut  of  the  rafter,  and  the  exact  length  required. 
It  will  be  noticed  that  the  square  has  been  applied  to  the 
timber  thirteen  times. 

The  reason  for  this  is,  that  the  building  is  twenty-six  feet 
wide,  the  half  of  which  is  thirteen  feet,  the  distance  that 
one  rafter  is  expected  to  reach,  so,  if  the  building  was  thirty 
feet  wide,  we  should  be  obliged  to  apply  the  square  fifteen 
times  instead  of  thirteen.  We  may  take  it  for  granted, 
then,  that  in  all  cases  where  this  method  is  employed  to 
obtain  the  lengths  and  bevels,  or  cuts  of  rafters,  we  must 
apply  the  square  half  as  many  times  as  there  are  feet  in  the 
width  of  the  building  being  covered.  If  the  roof  to  be 
covered  is  one-third  pitch,  all  to  be  done  is  to  take  12 
inches  on  one  side  of  the  square  and  8 inches  on  the  other, 
and  operate  as  for  quarter  pitch. 

We  shall  frequently  meet  with  roofs  much  more  acute 
than  the  ones  shown,  but  it  will  be  easy  to  see  how  they 
can  be  managed.  For  instance,  where  the  rafters  are  at 
right-angles  to  each  other,  apply  the  square  the  same  as 
for  braces  of  equal  run,  that  is  to  say,  keep  the  12  mark  on 
the  blade,  and  the  12  mark  on  the  tongue,  on  the  working 
line.  When  a roof  is  more  acute,  or  “ steeper”  than  a 
right-angle,  take  a greater  figure  than  twelve  on  one  side 
of  the  square,  and  twelve  on  the  other. 


and  its  uses. 


23 


Whenever  a drawing  of  a roof  is  to  be  followed,  we  can 
soon  find  out  how  to  employ  the  square,  by  laying  it  on 
the  drawing,  as  shown  in  Fig.  12.  Of  course,  something 
depends  on  the  scale  to  which  the  drawing  is  made.  If 
any  of  the  ordinary  fractions 
of  an  inch  are  used,  the  intelli- 
gent workman  will  have  no 
difficulty  in  discovering  what 
figures  to  make  use  of  to  get 
the  “cuts”  and  length  de- 
sired. 

Sometimes  there  may  be  a 
fraction  of  a foot  in  this  divis- 
ion ; when  such  is  the  case,  it 
can  be  dealt  with  as  follows  : 
suppose  there  is  a fraction  of 
a foot,  say  8 inches,  the  half 
of  which  would  be  4 inches, 
or  ^ of  a foot ; then,  if  the 
roof  is  quarter  pitch,  all  to  be 
done  is  to  place  the  square, 
with  the  4 inch  mark  on  the 
blade,  and  the  2 inch  mark  on 
the  tongue,  on  the  centre  line 
of  the  rafter,  and  the  distance 
between  .these  points  is  the 
extra  length  required,  and  the  line  down  the  tongue  is  the 
bevel  at  the  point  of  the  rafter.  On  Fig.  13,  is  shown 
an  application  of  this  method.  All  other  pitches  and  frac- 
tions can  be  treated  in  this  manner  without  overtaxing  the, 
ingenuity  of  the  workman. 


24 


THE  STEEL  SQUARE 


Fig.  14. 


AND  ITS  USES. 


*5 


Sufficient  has  been  shown  to  enable  the  student,  if  he 
has  mastered  it,  to  find  the  lengths  and  bevels  of  any  com- 
mon rafter ; therefore,  for  the  present,  we  will  leave  saddle 
roofs,  and  try  what  can  be  done  with  the  square  in  de- 
termining the  lengths  and  bevels  of  “ hips,”  valleys,  and 
cripples. 


Fig.  t5- 


Fig.  14  shows  how  to  get  bevels  on  the  top  end  of  vertical 
boarding,  at  the  gable  ends,  suitable  for  the  quarter  pitch, 
at  Fig.  10. 

At  Fig.  15,  is  shown  a method  for  finding  the  bevel  for 
horizontal  boarding,  collar  ties,  etc. 

Hip  Rafters. — Fig.  16,  is  supposed  to  be  the  pitch  of  a 
roof  furnished  by  an  architect,  with  the  square  applied  to 
the  pitch.  The  end  of  the  long  blade  must  only  just  enter 


2 6 


THE  STEEL  SQUARE 


and  its  USES. 


27 


the  fence,  as  shown  in  the  drawing,  and  the  short  end  must 
be  adjusted  to  the  pitch  of  the  roof,  whatever  it  may  be. 
Fig.  17  shows  the  square  set  to  the  pitch  of  the  hip  rafter. 
The  squares  as  set  give  the  plumb  and  level  cuts.  Fig.  18 
is -the  rafter  plan  of  a house  18  by  24  feet;  the  rafters  are 
laid  off  on  the  level,  and  measure  nine  feet  from  centre  of 
ridge  to  outside  of  wall ; there  should  be  a rafter  pattern 
with  a plumb  cut  at  one  end,  and  the  foot  cut  at  the  other, 
got  out  as  previously  shown.  (Figs.  16, 17, 18,  P.)  When  the 
rafter  foot  is  marked,  place  the  end  of  the  long  blade  of  the 
square  to  the  wall  line,  as  in  drawing,  and  mark  across  the 
rafter  at  the  outside  of  the  short  blade,  and  these  marks  on 
the  rafter  pitch  will  correspond  with  two  feet  on  the  level 
plan , slide  the  square  up  the  rafter  and  place  the  end  of  the 
long  blade  to  the  mark  last  made,  and  mark  outside  the  short 
blade  as  before,  repeat  the  application  until  nine  feet  are 
measured  off,  and  then  the  length  of  the  rafter  is  correct ; 
remember  to  mark  off  one-half  the  thickness  of  ridge-piece. 
The  rafters  are  laid  off  on  part  of  plan  to  show  the  appearance 
of  the  rafters  in  a roof  of  this  kind,  but  for  working  purposes 
the  rafters  1,  2,  3,  4,  5,  and  6,  with  one  hip  rafter,  is  all 
that  is  required. 

Hip-roof  Framing. — We  first  lay  off*  common  rafter, 
which  has  been  previously  explained ; but  deeming  it  ne- 
cessary to  give  a formula  in  figures  to  avoid  making  a plan, 
we  take  }%  pitch.  This  pitch  is  the  width  of  building,  to 
point  of  rafter  from  wall  plate  or  base.  For  an  example, 
always  use  8,  which  is  l/z  of  24,  on  tongues  for  altitude;  12, 
y2  the  width  of  24,  on  blade  for  base.  This  cuts  common 
rafter.  Next  is  the  hip-rafter.  It  must  be  understood  that 


2 


THE  STEEL  SQUARE 


the  diagonal  of  12  and  12  is  17  in  framing,  and  the  hip  is 
the  diagonal  of  a square  added  to  the  rise  of  roof ; there- 
fore we  take  8 on  tongue  and  1 7 on  blade ; run  the  same 
number  of  times  as  common  rafter  (rule  to  find  distance 
of  hip  diagonal  a2  + a2  -f  b2  = y2).  To  cut  jack  rafters,  divide 
the  numbers  of  openings  for  common  rafter.  Suppose  we 
have  5 jacks,  with  six  openings,  our  common  rafter  12  feet 
long,  each  jack  would  be  2 feet  shorter.  First  10  feet, 
second  8 feet,  third  6 feet,  and  so  on.  The  top  down  cut 
the  same  as  cut  6f  common  rafter ; foot  also  the  same. 
To  cut  mitre  to  fit  hip.  Take  half  the  width  of  building 
on  tongue  and  length  of  common  rafter  on  blade ; blade 
gives  cut.  Now  find  the  diagonal  of  8 and  12,  which 
is  1442,  call  it  14  7-16,  take  12  on  tongue,  14  7-16  on 
blade;  blade  gives  cut.  The  hip-rafter  must  be  beveled  to 
suit  jacks;  height  of  hip  on  tongue,  length  of  hip  on  blade; 
tongue  gives  bevel.  Then  we  take  8 on  tongue  18^  on 
blade  ; tongue  gives  the  bevel.  Those  figures  will  span  all 
cuts  in  putting  on  cornice  and  sheathing.  To  cut  bed 
moulds  for  gable  to  fit  under  cornice,  take  half  width  of 
building  on -tongue  length  of  common  rafter  on  blade; 
blade  gives  cut ; machine  mouldings  will  not  member,  but 
this  gives  a solid  joint ; and  to  member  properly  it  is  neces- 
sary to  make  Moulding  by  hand,  the  diagonal  plumb  cut 
differences.  I find  a great  many  mechanics  puzzled  to 
makes  the  cuts  for  a valley.  To  cut  planceer,  to  run  up 
valley,  take  heighth  of  rafter  on  tongue,  length  of  rafter  on 
blade  ; tongue  gives  cut.  The  plumb  cut  takes  the  height 
of  hip-rafter  on  tongue,  length  of  hip-rafter  on  blade; 
tongue  gives  cut.  These  figures  give  the  cuts  for  yi  pitch 
only,  regardless  of  width  of  building. 


AND  ITS  USES. 


29 


For  a hopper  the  mitre  is  cut  on  the  same  principle. 
To  make  a butt  joint,  take  the  width  of  side  on  blade,  and 
half  the  flare  on  tongue;  the  latter  gives  the  cut.  You 
will  observe  that  a hip-roof  is  the  same  as  a hopper  in- 
verted. The  cuts  for  the  edges  of  the  pieces  of  a hexagonal 
hopper  are  found  this  way : Subtract  the  width  of  one 
piece  at  the  bottom  from  the  width  of  same  at  top,  take 
remainder  on  tongue,  depth  of  side  on  blade ; tongue  gives 
the  cut.  The  cut  on  the  face  of  sides  : Take  7-12  of  the 
rise  on  tongues  and  the  depth  of  side  on  blade ; tongue 
gives  cut.  The  bevel  of  top  and  bottom  : Take  rise  on 
blade,  run  on  tongue ; tongue  gives  cut. 

Fig.  19  exhibits  two  methods  of  finding  the  “ backing  ” 
of  the  angle  on  hip-rafter.  The  methods  are  as  simple  as 
any  known.  Take  the  length  of  the  rafter  on  the  blade, 
and  the  rise  on  the  short  blade  or  tongue,  place  the 
square  on  the  line  D E,  the  plan  of  the  hip,  the  angle  is 
given  to  bevel  the  hip-rafter,  as  shown  at  F.  This  method 
gives  the  angle,  only  for  a right-angled  plan,  where  the 
pitches  are  the  same,  and  ?io  other . 

The  other  method  applies  to  right,  obtuse,  and  acute 
angles,  where  the  pitches  are  the  same.  At  the  angle  d 
will  be  seen  the  line  from  the  points  k l,  at  the  intersec- 
tion of  the  sides  of  the  angle  rafter  with  the  sides  of  the 
plan. 

With  one  point  of  the  compass  at  d,  describe  the  curve 
from  the  line  as  shown.  Tangential  to  the  curve  draw 
the  dotted  line,  cutting  a,  then  draw  a line  parallel  to  a b, 
the  pitch  of  the  hip.  The  piten  or  bevel,  will  be  found 
at  G,  which  is  a section  of  the  hip-rafter. 

This  problem  is  taken  from  “ Gould’s  Carpenters’ 


3° 


THE  STEEL  SQUARE 


Fig.  19. 


AND  ITS  USES. 


31 


Guide,”  but  has  been  in  practice  among  workmen  for 
many  years. 


Fig.  20  exhibits  a method  of  finding  the  cuts  in  a mitre 
box,  by  placing  the  square  on  the  line  a b at  equal  dis- 
tances from  the  heel  of  the  square,  say  ten  inches.  The 
bevel  is  shown  to  prove  the  truth  of  the  lines  by  applying 
it  to  opposite  sides  of  the  square. 

Stairs. — In  laying  out  stairs  with  the  square,  it  is  neces- 
sary to  first  determine  the  height  from  the  top  of  the  floor 
on  which  the  stairs  start  from,  to  the  floor  on  which  they 
are  to  land  ; also  the  “ run  ” or  the  distance  of  their  hori- 
zontal stretch.  These  lengths  being  obtained,  the  rest  is 
easy. 

Fig.  21  shows  a part  of  a stair  string,  with  the  “ square  ” 
laid  on,  showing  its  application  in  cutting  out  a pitch-board; 
As  the  square  is  placed  it  shows  10  inches  for  the  tread  and 
7 inches  for  the  rise. 

To  cut  a pitch-board,  after  the  tread  and  rise  have  been 


37 


THE  STEEL  SQUARE 


determined,  proceed  as  follows:  Take  a piece  of  thin,  clear 
stuff,  and  lay  the  square  on  the  face  edge,  as  shown  in  the 
figure,  and  mark  out  the  pitch-board  with  a sharp  knife; 


then  cut  out  with  a fine  saw  and  dress  to  knife  marks,  nail 
a piece  on  the  longest  edge  of  the  pitch-board  for  a fence, 
and  it  is  ready  for  use. 

Fig.  22  is  a.  rod,  with  the  number  and  heighth  of  steps 
for  a rough  flight  of  stairs  to  lead  down  into  a cellar  or 
elsewhere. 

Fig.  ’23  is  a step-ladder,  sufficiently  inclined  to  permit  3 
person  to  pass  up  and  down  on  it  with  convenience.  To 
lay  off  the  treads,  level  across  the  pitch  of  the  ladder,  set 
the  short  side  of  the  square  on  the  floor,  at  the  foot  of  the 
string,  after  the  string  is  cut,  to  fit  the  floor  and  trimmer 
joists.  Fasten  the  fence  on  the  square,  as  shown  at  Fig.  5. 
The  height  of  the  steps  in  this  case  is  nine  inches,  so  it  will 
be  seen  that  it  is  an  easy  matter  to  lay  off  the  string,  as  the 


CELLER  STEPS 


AND  ITS  USES. 


33 


TOP  OF  JOIST 


34 


THE  STEEL  SQUARE 


long  side  of  the  square  hangs  plumb,  and  nine  inches  up 
its  length  will  be  the  distance  from  one  step  to  the  next 
one. 

Fig.  24  shows  the  square  and  fence  in  position  on  the 
string. 

The  opening  in  the  floor  at  the  top  of  the  string  shows 
the  ends  of  trimming  joists,  five  feet  apart. 

Fig.  25  shows  how  to  divide  a board  into  an  even  number 
of  parts,  each  part  being  equal,  when  the  same  is  an  un- 
even number  of  inches,  or  parts  of  an  inch  in  width.  Lay 
the  square  as  shown,  with  the  ends  of  the  square  on  the 
edges  of  the  board,  then  the  points  of  division  will  be 
found  at  6,  12,  and  18,  for  dividing  the  board  in  four 
equal  parts;  or  at  4,  8,  12,  16,  and  20,  if  it  is  desired  to 
divide  the  board  into  six  equal  parts.  Of  course,  the 
common  two-foot  rule  will  answer  this  purpose  as  well 
as  the  square,  but  it  is  not  always  convenient. 

Fig.  26  shows  how  a circle  can  be  described  by  means 
of  a “ steel  square  ” without  having  recourse  to  its  centre. 

At  the  extremities  of  the  diameter,  a,  o,  fix  two  pins,  as 
shown ; then  by  sliding  the  sides  of  the  square  in  contact 
with  the  pins,  and  holding  a pencil  at  the  point  x,  a semi- 
circle will  be  struck.  Reverse  the  square,  repeat  the  pro 
cess,  and  the  circle  is  complete. 

Miscellaneous  Rules  — The  following  rules  have  been 
tested  over  and  over  again  by  the  writer,  and  found  reliable 
in  every  instance.  They  have  been  known  to  advanced 
workmen  for  many  years,  but  were  never  published,  so  far 
as  the  writer  knows,  until  they  appeared  in  th z Builder  and 
Wood - Worker , some  years  ago : 


AND  ITS  USES. 


35 


Measurement. — Let  us  suppose  that  we  have  a pile  of 
lumber  to  measure,  the  boards  being  of  different  widths,  and 
say  1 6 feet  long.  We  take  our  square  and  a bevel  with  a 
long  blade  and  proceed  as  follows  : First  we  set  the  bevel 
at  1 2 inches  on  the  tongue  of  the  square,  because  we  want 
to'  find  the  contents  of  the  board  in  feet,  1 2 inches  being 
one  foot ; now  we  set  the  other  end  of  the  bevel  blade  on  the 
1 6 inch  mark  on  the  blade  of  the  square,  because  the  boards 
are  16  feet  long.  Now,  it  must  be  quite  evident  to  any 
one  who  would  think  for  a moment,  that  a board  1 2 inches, 
or  one  foot  wide,  and  16  feet  long,  must  contain  16  feet  of 
lumber.  Very  well,  then  we  have  16,  the  length,  on  the 
blade.  Now,  we  have  a board  11  inches  wide,  we  just 
move  our  bevel  from  the  1 2 inch  mark  to  the  1 1 inch  mark, 
and  look  on  the  blade  of  the  square  for  the  true  answer ; 
and  so  on  with  any  width,  so  long  as  the  stuff  is  16  feet 
long.  If  the  stuff  is  2 inches  thick,  double  the  answer,  if 
3 inches  thick,  treble  the  answer,  etc. 

Now,  if  we  have  stuff  14  feet  long,  we  simply  change 
the  bevel  blade  from  16  inches  on  the  square  blade,  to  14 
inches,  keeping  the  other  end  of  the  bevel  on  the  12  inch 
mark,  12  inches  being  the  constant  figure  on  that  side  of 
the  square,  and  it  will  easily  be  seen  that  any  length  of 
stuff  within  the  range  of  the  square  can  be  measured  ac- 
curately by  this  method. 

If  we  want  to  find  out  how  many  yards  of  plastering  or 
painting  there  are  in  a wall,  it  can  be  done  by  this  method 
quite  easily.  Let  us  suppose  a wall  to  be  12  feet  high  and 
18  feet  long,  and  we  want  to  find  out  how  many  yards  of 
plastering  or  painting  there  are  in  it,  we  set  the  bevel  on 
the  9 inch  mark  on  the  tongue  (we  take  9 inches  because  9 


36 


THE  STEEL  SQUARE 


square  feet  make  one  square  yard,)  we  take  18  inches,  one 
of  the  dimensions  of  the  wall,  on  the  blade  of  the  square ; 
then  after  screwing  the  bevel  tight,  we  slide  it  from  9 inches 
to  12  inches,  the  latter  number  being  the  other  dimension, 
and  the  answer  will  be  found  on  the  blade  of  the  square. 
It  must  be  understood  that  9 inches  must  be  a constant 
figure  when  you  want  the  answer  to  be  in  yards,  and  in 
measuring  for  plastering  it  is  as  well  to  set  the  other  end  of 
the  bevel  'on  the  figure  that  corresponds  with  the  height  of 
the  ceiling,  and  then  there  will  require  no  movement  of 
the  bevel  further  than  to  place  it  on  the  third  dimension. 
This  last  rule  is  a very  simple  and  very  useful  one ; of  course 
“ openings  ” will  have  to  be  allowed  for,  as  this  rule  gives 
the  whole  measurement. 

If  the  diagonal  of  any  parallelogram  within  the  range  of 
the  square  is  required,  it  can  be  obtained  as  follows : Set  the 
blade  of  the  bevel  on  8^  in.  on  the  tongue  of  the  square, 
and  at  1 2^4  in.  on  the  blade ; securely  fasten  the  bevel  at  this 
angle.  Now,  suppose  the  parallelogram  or  square  to  be  11 
inches  on  the  side,  then  move  the  bevel  to  the  1 1 inch  mark 
on  the  tongue  oj  the  square,  and  the  answer,  15  9-16,  will  be 
found  on  the  blade.  All  problems  of  this  nature  can  be  solved 
with  the  square  and  bevel  as  the  latter  is  now  set.  There 
is  no  particular  reason  for  using  8^  and  12^4,  only  that 
they  are  in  exact  proportion  to  70  and  99.  4^4  and  6 

3-16  would  do  just  as  well,  but  would  not  admit  as  ready 
an  adjustment  of  the  bevel. 

To  find  the  circumference  of  a circle  with  the  square  and 
bevel  proceed  as  follows : Set  the  bevel  to  7 on  the  tongue 

and  22  on  the  blade;  move  the  bevel  to  the  given  diameter 
on  the  tongue  of  the  square,  and  the  approximate  answer 


AND  ITS  USES. 


37 


will  be  found  on  the  blade.  When  the  circumference  is 
wanted  the  operation  is  simply  reversed,  that  is,  we  put 
the  bevel  on  the  blade  and  look  on  the  tongue  of  the  square 
for  the  answer. 

If  we  want  to  find  the  side  of  the  greatest  square  that 
can  be  inscribed  in  a given  circle,  when  the  diameter  is 
given,  we  set  the  bevel  to  8l/>  on  the  tongue  and  1 2 on  the 
blade.  Then  set  the  bevel  of  the  diameter,  on  the  blade, 
and  the  answer  will  be  found  on  the  tongue. 

The  circumference  of  an  ellipse  or  oval  is  found  by  set- 
ting 5^4  inches  on  the  tongue  and  8^  inches  on  the  blade  ; 
then  set  the  bevel  to  the  sum  of  the  longest  and  shortest 
diameters  on  the  tongue,  and  the  blade  gives  the  answer. 

To  find  a square  of  equal  area  to  a given  circle,  we  set  the 
bevel  to  9 ^ inches  on  the  tongue,  and  1 1 inches  on  the  blade ; 
then  move  the  bevel  to  the  diameter  of  the  circle  on  the 
blade,  and  the  answer  will  be  found  on  the  tongue.  If  the 
circumference  of  the  circle  is  given,  and  we  want  to  find  a 
square  containing  the  same  area,  we  set  the  bevel  to  5 *4 
inches  on  the  tongue  and  19^  inches  on  the  blade. 

On  Fig.  27  is  shown  a method  to  determine  the  pro- 
portions of  any  circular  presses  or  other  cylinderical  bodies, 
by  the  use  of  the  square.  Suppose  the  small  circle,  n,  to 
be  five  inches*  in  diameter  and  the  circle  r is  ten  inches  in 
diameter,  and  it  is  required  to  make  another  circle,  z,  to 
contain  the  same  area  as  the  two  circles  n and  r.  Meas- 
ure line  a , on  the  square  d,  from  five  on  the  tongue  to  10 
on  the  blade,  and  the  length  of  this  line  a from  the  two 
points  named  will  be  the  diameter,  of  the  larger  circle  z. 
And  again,  if  you  want  to  run  these  circles  into  a fourth 
one,  set  the  diameter  of  the  third  on  the  tongue  of  the  souare, 


38 


THE  STEEL  SQUARE 


and  the  diameter  of  z on  the  blade,  and  the  diagonal 
will  give  the  diameter  of  the  fourth  or  largest  circle,  and  the 
same  rule  may  be  carried  out  to  infinite  extent.  The  rule 
is  reversed  by  taking  the  diameter  of  the  greater  circle  and 
laying  diagonally  on  the  square,  and  letting  the  ends  touch 


vhatever  points  on  the  outside  edge  of  the  square.  These 
points  will  give  the  diameter  of  two  circles,  which  com- 
bined, will  contain  the  same  area  as  the  larger  circle.  The 
same  rule  can  also  be  applied  to  squares,  cubes,  triangles, 
rectangles,  and  all  other  regular  figures,  by  taking  similar 
dimensions  only ; that  is,  if  the  largest  side  of  one  triangle 
is  taken,  the  largest  side  of  the  other  must  also  be  taken, 
and  the  result  will  be  the  largest  side  of  the  required  tri- 
angle, and  so  with  the  shortest  side. 

In  Fig.  28  we  show  how  the  centre  of  a circle  may  be  de- 
termined without  the  use  of  compasses ; this  is  based  on 
the  principle  that  a circle  can  be  drawn  through  any  three 
points  that  are  not  actually  in  a straight  line.  Suppose  we 
take  a b c d for  four  given  points,  then  draw  a line  from  a 


AND  ITS  USES. 


39 


to  d,  and  from  b to  c ; get  the  centre  of  these  lines,  and 
square  from  these  centres  as  shown,  and  when  the  square 
crosses,  the  line,  or  where  the  lines  intersect,  as  at  x,  there 
will  be  the  centre  of  the  circle.  This  is  a very  useful  rule,  and 


Fig.  ?8. 


by  keeping  it  in  mind  the  mechanic  may  very  frequently 
save  himself  much  trouble,  as  it  often  happens  that  it  is  ne- 
cessary to  find  the  centre  of  the  circle,  when  the  compasses 
are  not  at  hand. 

In  Fig.  29  we  show  how  the  square  can  be  used,  in  lieu 
of  the  trammel,  for  the  production  of  ellipses.  Here  the 
square,  e d f,  is  used  to  form  the  elliptical  quadrant, 
a b,  instead  of  the  cross  of  the  trammel ; h l k may  be 
simply  pins,  which  can  be  pressed  against  the  sides  of  the 
square  while  the  tracer  is  moved.  In  this  case  the  adjust- 
ment is  obtained  by  making  the  distance,  h l,  equal  to  the 
semi-axis  minor,  and  the  distance  l k,  equal  to  the  semi-axis 
major. 


40 


THE  STEEL  SQUARE 


Fig.  30  shows  a method  of  describing  a parabola  by 
means  of  a straight  rule  and  a square,  its  double  ordinate 
and  abscissa  being  given.  Let  a c be  the  double  ordinate, 
and  d b the  abscissa.  Bisect  d c in  f;  join  b f,  and  draw 
f e perpendicular  to  b f,  cutting  the  axis  b d produced  in 
£ From  b set  off  b g equal  to  d e,  and  g will  be  the  focus 
of  the  parabola.  Make  b l equal  to  b g,  and  lay  the  rule 
on  straight-edge  h k on  l,  and  parallel  to  a c.  Take  a 
string,  m f o,  equal  in  length  toLE;  attach  one  of  its  ends 
to  a pin,  or  other  fastening,  at  g,  and  its  other  end  to  the 
end  m,  of  the  square  m n o.  If  now  the  square  be  slid 
along  the  straight-edge,  and  the  string  be  pressed  against 


AND  ITS  USES. 


4* 

its  edge  m n,  a pencil  placed  in  the  bight  at  f will  describe 
the  curve. 


The  two  arms  of  a horizontal  lever  are  respectively 
9 inches  and  13  inches  in  length  from  the  suspending 
point;  a weight  of  10  lbs.  is  suspended  from  the  shorter 
arm,  and  it  is  required  to  know  what  weight  will  be  re- 
quired to  suspend  on  the  long  arm  to  make  it  balance. 
Set  a bevel  on  the  blade  of  square  at  13  inches  and  the 
other  end  of  the  bevel  on  the  9 inch  mark  on  tongue  of 
square,  then  slide  the  bevel  from  13  inches  to  10  on  the 
blade  of  square,  and  the  answer  will  be  found  on  the 
tongue  of  the  square.  It  is  easy  to  see  how  this  rule  can 
be  reversed  so  that  a weight  required  for  the  shorter  arm 
can  be  found. 

Fig.  31  shows  how  to  get  the  flare  for  a hopper  4 feet 
across  the  top  and  16  inches  perpendicular  depth.  Add  to 
the  depth  one-third  of  the  required  size  of  the  discharge 


12 


THE  STEEL  SQUARE 


HALF  OF  DISCHARGE^  3"^ 


Fig.  31. 


hole  (the  draft  represents  a 6-inch  hole),  which  makes  18 
inches,  which  is  represented  on  the  tongue  of  the  square. 
(The  figures  on  the  draft  are  9 and  12,  which  produce  the 
same  bevel.)  Then  take  one-half,  24  inches  of  the  width 
across  the  top  of  the  hopper,  which  is  represented  on  the 
blade  of  the  square.  Than  scribe  along  the  blade  as  rep- 
resented by  the  dotted  lines,  which  gives  the  required  flare. 
(The  one-third  added  to  the  depth  is  near  enough  for 
all  practical  purpose  for  the  discharge.) 


Fig.  32. 


AND  ITS  USES* 


43 


Fig.  3 2 shows  how  to  apply  the  square  to  the  edge  of 
a board  in  order  to  obtain  the  bevel  to  form  the  joint. 
Using  the  same  figures  as  in  Fig.  31,  scribe  across  the  edge 
of  the  board  by  the  side  of  the  tongue,  as  shown  by  dotted 
lines.  The  long  point  being  the  outside. 


Fig.  33. 


On  Fig.  33  we  show  a quick  method  of  finding  the 
centre  of  a circle : Let  n n,  the  corner  of  the  square,  touch 
the  circumference,  and  where  the  blade  and  tongue  cross 
it  will  be  divided  equally ; then  move  the  square  to  any 
other  place  and  mark  in  the  same  way  and  straight  edge 
across,  and  where  the  line  crosses  a,  b,  as  at  o,  there  will 
be  the  centre  of  the  circle. 


44 


THE  STEEL  SQUARE 


i and  2,  Fig.  34,  are  taken  from  Gould’s  Wood-  Work - 
ing  Guide . 

The  portion  marked  a,  exhibits  a method  of  finding  the 
lines  for  eight-squaring  a piece  of  timber  with  the  square, 


by  placing  the  block  on  the 
piece,  and  making  the  points 
seven  inches  from  the  ends 
of  the  square,  from  which  to 
draw  the  lines  for  the  sides 
of  the  octagonal  piece  re- 
quired. 

At  the  heel  of  the  square 
is  shown  a method  of  cut- 
ting a board  to  fit  any  angle 
jr  with  the  square  and  compass, 
i by  placing  the  square  in  the 
angle,  and  taking  the  distance 
from  the  heel  of  the  square 
to  the  angle  a,  in  the  com- 
pass ; then  lay  the  square  on 
the  piece  to  be  fitted,  with 
the  distance  taken,  and  from 
the  point  a,  draw  the  line  a 
b,  which  will  give  the  angle 
to  cut  the  piece  required. 

At  2 is  shown  a method 
of  constructing  a polygonal 
figure  of  eight  sides ; by  placing  the  square  on  the  line  a b, 
with  equal  distances  on  the  blade  and  tongue,  as  shown ; 
the  curve  lines  show  the  method  of  transferring  the  dis- 
tances; the  diagonal  gives  the  intersection  at  the  angles. 


AND  ITS  USES. 


45 


There  are  at  least  a dozen  different  ways  of  forming  oc- 
tagonal figures  by  the  square ; some  of  them  are  tedious 
and  difficult,  while  others  can  not  be  applied  under  all  cir- 
cumstances. The  method  shown  at  Fig.  35  is  handy  and 
easily  understood. 


E»g.  35- 

An  equilateral  triangle  can  be  formed  by  taking  half  of 
one  side  on  the  tongue  of  the  square,  as  shown  at  Fig.  36. 
The  line  along  the  edge  of  the  tongue  forms  the  mitre  for 


the  triangle,  and  the  line  along  the  edge  of  the  blade  forms 
the  mitre  cut  for  the  joints  of  a hexagon,  and  as  six  equi- 


46 


THE  STEEL  SQUARE 


lateral  triangles  form  a hexagon  when  one  point  of  each  is 
placed  at  a central  point,  <?,  it  follows  that  a hexagon  may 
be  constructed  by  the  square  above. 

The  following  is  a good  method  for  obtaining  the  cuts 
for  a horizontal  and  raking  cornice ; it  is  correct  and  simple ; 
the  gutter  to  be  always  cut  a square  mitre. 

The  seat  or  run  of  the  rafter  on  the  blade,  r c,  Fig.  37, 
the  rise  of  the  roof  on  the  tongue,  a c,  mark  against  the 
tongue,  gives  the  cut  for  the  side  of  the  box,  a c.  The 


diagonal  a,  r,  which  is  the  length  of  the  rafter  on  the  blade 
a,  d,  the  seat  of  the  ralter  on  the  tongue  d,  s,  mark  against 
the  blade  gives  the  cut  across  the  box,  ad.  d a c is  the 
mitre  cut  to  fit  the  gutter;  then  if  we.  square  across  the 
box  from  a,  it  gives  f,  a,  c the  cut  for  the  gable  peak. 

At  Fig.  38  is  shown  a method  for  obtaining  either  the 
butt  or  mitre  cuts,  for  “ Hopper  ” work. 

The  line,  s s,  in  the  cut  represents  the  edge  of  a board ; 
the  line,  a b,  the  flare  of  hopper.  Lay  the  square  on  the 
face  of  the  board  so  that  the  blade  will  coincide  with  flare 
of  hopper,  a b,  then  mark  by  the  tongue  the  line  b c,  then 
square  from  edge  of  board,  s s,  cutting  the  angle  b. 

Now  we  have  a figure  that  will,  when  used  on  the  steel 


AND  ITS  USE^. 


47 


square,  give  the  cuts  for  a hopper  of  any  flare,  either  with 
butt  or  mitre  joints. 

To  find  bevel  to  cut  across  face 'of  board: 

Take  a b on  blade  and  a d on  tongue,  bevel  of  tongue 
is  the  bevel  required. 


Fig.  38: 


To  find  the  bevel  for  butt  joint : Take  b c on  blade 

and  a d on  tongue;  bevel  of  tongue  is  the  bevel  required. 

To  find  the  bevel  for  mitre  joint : Take  b c on  blade 

and  d c on  tongue ; bevel  of  tongue  is  the  bevel  required. 

It  will  be  seen  that  this  is  a very  simple  method  of 
solving  what  is  usually  considered  a very  difficult  problem. 

The  foregoing  are  only  a few  of  the  uses  to  which  the 
“ steel  square  ” may  be  applied,  as  the  student  will  soon 
discover  for  himself;  but  we  trust  the  young  mechanic  will 
not  rest  at  this  point,  but  will  pursue  his  studies  further,  as 
the  Author  can  assure  him  that  there  is  a large  field  yet 
undeveloped  where  he  will  find  it  instructive  and  profitable 
to  explore.  In  order  that  a more  thorough  knowledge  of 
this  very  useful  tool  may  be  acquired,  than  the  present 


48 


THE  STEEL  SQUARE 


manual  affords,  we  would  refer  the  reader  to  the  larger 
work  on  the  same  subject — i.  e,,  “ The  Steel  Square  and 
Its  Uses,”  a new  and  enlarged  edition  of  which  will  soon 
be  issued,  and  will  sell  at  about  one  dollar. 

The  writer  of  this  little  book  would  like  nothing  better 
than  to  see  copies  of  this  and  the  larger  work  on  the  same 
subject  in  the  hands  of  every  man  in  America,  who  in  any 
way  has  to  frequently  or  occasionally  use  the  “ Steel 
Square.”  The  study  of  the  subject  would  surely  make  a 
wiser,  and  consequently  a better,  citizen  of  the  man  who 
makes  it 


